Accelerometers, gyroscopes (spinning mass, optical, vibratory), IMUs, and error characteristics.
An INS needs six sensors: three accelerometers (measuring specific force along x, y, z) and three gyroscopes (measuring angular rate about x, y, z). Together they form an inertial measurement unit (IMU).
Accelerometer designs: pendulous (proof mass on a hinge, force-feedback) and vibrating-beam (frequency shifts with applied force).
Gyroscope designs: spinning mass (conservation of angular momentum), optical (ring laser or fiber-optic, Sagnac effect), and vibratory (Coriolis acceleration of a vibrating element, including MEMS).
A proof mass hangs from a hinge inside a case. When the case accelerates along the sensitive axis (perpendicular to the pendulous arm), the proof mass lags behind, deflecting.
Open-loop: springs balance the deflection; a pickoff reads the displacement. Simple but limited by spring nonlinearity, hysteresis, and sensitive-axis misalignment as the pendulum swings.
Closed-loop (force-feedback): an electromagnetic torquer holds the proof mass at its null position. The torquer current is proportional to specific force. Much better linearity and dynamic range. This is the precision standard.
MEMS pendulous accelerometers use electrostatic (not magnetic) torquers and capacitive or resistive pickoffs. They're tiny but noisier.
The VBA replaces the spring with a vibrating beam. The beam is driven at its resonant frequency. When compressed (by acceleration), the frequency decreases; when stretched, it increases. Measuring the frequency gives specific force.
Dual-beam designs (one compressed, one stretched) cancel common-mode errors and double the sensitivity. Quartz elements give the sharpest resonance peak. MEMS VBAs exist in both quartz and silicon.
The VBA is inherently open-loop but avoids the problems of springs: the proof mass barely moves, so the sensitive axis stays aligned with the case.
A spinning disc has angular momentum h along its spin axis. Apply a torque τ about an orthogonal axis, and the disc precesses about the mutually perpendicular axis: τ = ωp × h.
In a single-degree-of-freedom gyro, the rotor is free to precess about the output axis. When the case rotates about the input axis, the case torques the rotor about the input axis, causing precession about the output axis. A torquer (closed-loop) or spring (open-loop) balances this precession, and the balancing torque is proportional to angular rate.
Light travels at a constant speed in an inertial frame. Send light both ways around a closed loop. If the loop rotates, one path gets longer and the other shorter — the Sagnac effect.
Ring laser gyro (RLG): a gas laser in a triangular or square cavity. Two counter-propagating beams. Rotation shifts their frequencies. The beat frequency is proportional to angular rate. Problem: at low rates, the beams "lock in" (synchronize). Solution: mechanical dithering.
Fiber-optic gyro (IFOG): broadband light through a long fiber-optic coil. Rotation introduces a phase difference between the two paths. Sensitivity scales with coil area × number of turns. More reliable than RLGs.
Drive an element (string, beam, ring, hemisphere) in simple harmonic motion. When the gyro rotates, Coriolis acceleration pushes the vibrating element perpendicular to both the vibration and rotation axes. The amplitude of this secondary vibration is proportional to angular rate.
Most MEMS gyros are vibratory. Quartz gives better performance than silicon. The exception: the hemispherical resonator gyro (HRG), which offers aviation-grade performance in a compact package and is popular for space applications.
An IMU packages the six sensors with a processor, temperature sensor, calibration store, and power supplies. The processor performs unit conversion, error compensation, and range checks.
Most IMUs output integrated measurements: "delta-v"s (υibb, specific force integrated over the sampling interval) and "delta-θ"s (αibb, angular rate integrated over the interval). Output rates: 100–1000 Hz.
Every inertial sensor has errors. Understanding them is essential for Kalman filter design.
| Error Type | Description | Effect |
|---|---|---|
| Bias | Constant offset in output | Accel bias → velocity drift. Gyro bias → attitude drift → position drift |
| Scale factor | Output proportional to true value but scaled wrong | Errors grow with dynamic range |
| Cross-coupling | Sensitivity to off-axis inputs | Axis misalignment errors |
| Random noise | White noise, random walk, flicker noise | Integration amplifies high-frequency noise |
| g-dependent bias | Gyro bias varies with applied specific force | Heading error during acceleration |
Turn-on bias varies each time the sensor starts. In-run bias varies slowly during operation. Only the in-run component matters for long missions after initial alignment.