Inertial Navigation Simulator: IMU Drift, Bias & Error Growth

simulator advanced ~12 min
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δx ≈ 480 m — position drift after 30 minutes

With 1 mg accelerometer bias and 5°/hr gyro drift, standalone INS accumulates about 480 m position error after 30 minutes. Heading error grows to 3° from the initial 0.5° plus drift.

Formula

δx ≈ ½ · b_a · g · t²  (position error from accel bias)
δψ(t) = δψ₀ + ω_drift · t  (heading error growth)
T_Schuler = 2π√(R_e / g) ≈ 84.4 min

Navigation Without External Signals

Inertial navigation is the only positioning technology that requires no external signals — no satellites, no radio beacons, no landmarks. By continuously measuring acceleration and rotation from a known starting point, an INS computes position through pure dead-reckoning. This self-contained nature makes INS indispensable for submarines, missiles, spacecraft, and any platform operating in GPS-denied environments.

The Sensor Triad

An inertial measurement unit (IMU) contains three orthogonal accelerometers and three orthogonal gyroscopes. The gyroscopes track the vehicle's orientation, allowing the navigation computer to transform accelerometer measurements from body frame to navigation frame. After subtracting the gravity vector, the remaining specific force is integrated to yield velocity, then integrated again to yield position. This double integration is both the method's power and its Achilles heel.

The Tyranny of Integration

Double integration amplifies every sensor error. A tiny accelerometer bias — just 1 milligravity (0.01 m/s²) — produces a position error that grows quadratically: 18 m after 1 minute, 1.8 km after 10 minutes, 29 km after 1 hour. Gyroscope drift is even more insidious: it misaligns the reference frame, coupling the enormous gravity vector (9.8 m/s²) into the horizontal channels. The simulation lets you watch these errors accumulate in real time.

Sensor Grades and Applications

INS accuracy spans orders of magnitude depending on sensor quality. Consumer MEMS IMUs ($5, 10 mg bias, 100°/hr drift) are usable only with continuous GPS aiding. Tactical-grade IMUs ($1k, 1 mg, 10°/hr) support minutes of GPS outage. Navigation-grade IMUs ($50k, 0.05 mg, 0.01°/hr) enable hours of standalone navigation for aircraft and ships. Strategic-grade IMUs ($500k+) guide ICBMs and submarines for weeks. The simulation reveals why this hierarchy exists.

FAQ

How does inertial navigation work?

An inertial navigation system (INS) uses accelerometers and gyroscopes to track position by dead-reckoning from a known starting point. Gyroscopes maintain orientation (attitude), while accelerometers measure specific force. After subtracting gravity, the remaining acceleration is integrated once for velocity and twice for position — all without external signals.

Why does INS error grow over time?

Any small constant bias in the accelerometer gets integrated twice, producing position error that grows as ½·b·t². A 1 mg bias (0.01 m/s²) produces 18 m error after 1 minute and 1.8 km after 10 minutes. Gyroscope drift compounds this by misaligning the reference frame, coupling gravity into horizontal channels.

What is the Schuler oscillation?

A navigation-grade INS exhibits a natural oscillation with period 84.4 minutes (the Schuler period = 2π√(R_e/g)). Errors in the vertical channel cause the platform to tilt, mixing gravity into horizontal acceleration, which creates a self-correcting oscillation. This fortuitous property limits horizontal velocity errors to bounded oscillations rather than unbounded drift.

What is the difference between strapdown and gimbaled INS?

Gimbaled systems mount sensors on a mechanically stabilized platform that maintains alignment with the navigation frame. Strapdown systems fix sensors rigidly to the vehicle and use computational attitude updates. Modern systems are almost universally strapdown — they are lighter, cheaper, more reliable, and computational power has eliminated their historical accuracy disadvantage.

Sources

Embed

<iframe src="https://homo-deus.com/lab/navigation/inertial-navigation/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub