Inertial Navigation

Inertial Navigation (gnc-autonomy/navigation/inertial-navigation)

Use when the task is inertial navigation: mechanization choice, error growth from sensor imperfections, Schuler tuning, alignment, and INS/GPS integration.

Domain quick reference

• An INS measures specific force and rotation rate with accelerometers and gyros and integrates them into position, velocity, and attitude. It needs no external reference while it runs; its errors grow with time.
• Two mechanizations: gimbaled keeps the sensor cluster on a stabilized platform driven by torque loops; strapdown mounts the cluster rigidly to the vehicle and carries the attitude in software (direction cosine matrix or quaternion). Strapdown is the standard on modern aircraft; gimbaled survives in some reference systems and heritage designs.
• Accelerometer errors: bias, scale factor, misalignment, random walk. Bias dominates the position error: double integration of a constant bias b gives velocity error bt and position error 0.5b*t^2 (short-time model).
• Gyro errors: bias (drift), scale factor, g-sensitive drift, angle random walk. A constant drift eps about a level axis tilts the computed vertical and couples gravity; the position error grows as (1/6)geps*t^3, the cubic term that limits an unaided INS.
• Schuler tuning: the level loop is tuned as a pendulum of length equal to the Earth radius, so the natural period is T = 2pisqrt(R/g) = 84.4 min. A Schuler-tuned INS does not diverge from leveling errors; they oscillate at the Schuler frequency, and a constant accelerometer bias settles to the bounded offset b*R/g (about 650 m per mg).
• Alignment: leveling with accelerometers (gravity defines the vertical) and gyrocompassing with the Earth-rate vector (7.2921159e-5 rad/s; the north component omega_e*cos(lat) defines north). Azimuth alignment takes minutes and is set by gyro noise.
• INS/GPS integration: the GPS solution bounds the unbounded INS errors, and the INS bridges GPS outages and smooths the output. Loosely coupled (position and velocity updates into the navigation filter), tightly coupled (raw pseudorange and carrier phase), or deeply coupled (aiding inside the receiver). GPS errors are bounded; INS errors are bounded only by aiding.
• Key numbers: g0 = 9.80665 m/s^2; mean Earth radius 6371 km; Schuler period 84.4 min; Earth rate 7.2921159e-5 rad/s; 1 deg/h = 4.848e-6 rad/s.
• ARINC 429 (reference-only) is the civil data bus over which inertial reference systems broadcast attitude, heading, position, and velocity; the standard is cited, never copied.

Workflow

1. Choose the mechanization: strapdown (software attitude) unless the design is a gimbaled reference system.
2. Convert the gyro drift with deg_per_hour_to_rad_s(deg_per_hour) before it enters any growth model.
3. Estimate the short-time bias error with accel_bias_position_error(bias, t) and the velocity error with accel_bias_velocity_error(bias, t).
4. Estimate the cubic gyro error with gyro_drift_position_error(deg_per_hour, t); this term usually sets the unaided error budget.
5. Check the loop tuning with schuler_period() and schuler_frequency(); confirm 84.4 min.
6. Bound the long-time bias error with schuler_steady_state_error(bias).
7. Resolve the Earth rate for gyrocompassing with earth_rate_component(lat_rad).
8. Accumulate gyro noise with angle_random_walk_sigma(arw_deg_per_sqrt_h, t_hours).
9. Scope the integration: loosely coupled when the receiver delivers a position and velocity solution; tighter coupling when raw measurements and a shared filter are available.

Pitfalls

• Double counting the bias error: 0.5bt^2 is the short-time free response; over tens of minutes the Schuler loop bends it into the bounded b*R/g offset. Quote both, and say which applies.
• Forgetting the cubic gyro term: at 0.001 deg/h it is about 370 m after one hour, at 0.01 deg/h about 3.7 km. Gyro drift, not accelerometer bias, usually sets the unaided budget.
• Mixing units: deg/h must convert to rad/s (1 deg/h = 4.848e-6 rad/s) before the growth formulas.
• Treating the Schuler period as a damping time: it is an oscillation period, not a settling time; errors oscillate at 84.4 min rather than decay.
• Confusing specific force with acceleration: accelerometers measure specific force, so gravity must be modeled and removed in the integration.
• Forgetting that azimuth alignment needs Earth-rate sensing: the north component shrinks as cos(lat), so gyrocompassing degrades at high latitude.

Behavior contract (gate 3)

The Schuler quantities, bias and drift error growth, steady-state offset, Earth-rate components, and random-walk accumulation are exercised by the gate 3 contract test: scripts/test_inertial_navigation.py against scripts/inertial_navigation_logic.py (stdlib unittest, offline). Run: python3 skills/gnc-autonomy/navigation/inertial-navigation/scripts/test_inertial_navigation.py

Compliance

• ARINC 429 is proprietary (ARINC/SAE ITC); inertial reference systems on civil aircraft broadcast over it. Name and paraphrase only per standards-map.yaml, reference-only: true.
• compliance: STANDARDS-REF, gated: false.