IMUs and magnetometers

Why this matters

The IMU (inertial measurement unit) is the heart of the flight controller. It contains gyroscopes, which measure rotation rate, and accelerometers, which measure acceleration. The controller reads the IMU hundreds to thousands of times per second to hold attitude. The magnetometer measures the Earth’s magnetic field to give heading. If these sensors are slightly off, the drone tilts, drifts or flies in the wrong direction.

MEMS and the error model

Almost all drone sensors are MEMS (micro-electro-mechanical systems): micrometre-scale mechanical structures on a silicon chip. An accelerometer has a small proof mass on springs; when it accelerates, the mass moves and a capacitance changes. A MEMS gyro has a mass that vibrates continuously; when it rotates, the Coriolis force makes it vibrate along another axis (Groves, 2013).

Each axis has a basic error model $\tilde{a}=sa+b+n$, where $s$ is the scale factor, $b$ the bias and $n$ the noise. Real sensors also have misaligned axes and temperature dependence. Datasheets give noise levels; TDK’s ICM-42688-P, for example, specifies gyro noise of 2.8 mdps/√Hz and accelerometer noise of 70 µg/√Hz.

Six-position accelerometer calibration

At rest, an accelerometer measures gravity with magnitude exactly $g$. Placing each axis pointing up and then down, six positions in total, gives $+g$ and $-g$ readings on every axis, from which bias and scale factor follow directly. The ArduPilot guide uses these six positions (level, right side, left side, nose down, nose up and on its back), and IEEE 1293-2018 describes tumble testing of accelerometers in detail.

import math

G = 9.80665
readings = {"x": (9.93, -9.69), "y": (9.85, -9.77), "z": (9.62, -10.02)}   # m/s² axis pointing up, pointing down

cal = {}
for axis, (up, down) in readings.items():
    bias = (up + down) / 2
    scale = (up - down) / (2 * G)
    cal[axis] = (bias, scale)
    print(f"{axis}: bias {bias:+.3f} m/s^2, scale factor {scale:.4f}")

raw = (0.12, 0.04, -10.02)            # truly level hover, z axis pointing down
fixed = [(r - cal[a][0]) / cal[a][1] for a, r in zip("xyz", raw)]
tilt = lambda v: math.degrees(math.atan2(math.hypot(v[0], v[1]), -v[2]))
print(f"apparent tilt before {tilt(raw):.2f} deg, after {tilt(fixed):.2f} deg")

x: bias +0.120 m/s^2, scale factor 1.0003
y: bias +0.040 m/s^2, scale factor 1.0003
z: bias -0.200 m/s^2, scale factor 1.0014
apparent tilt before 0.72 deg, after 0.00 deg

Allan deviation

Recording at rest for a long time and computing the Allan deviation against averaging time $\tau$ separates noise types. NXP application note AN5087 and El-Sheimy et al. (2008) explain how to read this log-log plot.

• Where the curve falls with slope −1/2, the noise is white; its value on this line at $\tau =1$ s is the angle random walk (ARW).
• The flat region before the curve rises is the bias instability, the smallest amount the bias wanders. AN5087 reads the minimum directly, while the IEEE gyro standard divides the minimum by 0.664; a report must say which method it uses.

These values set the noise parameters of the EKF and help compare two sensor models before buying.

Magnetometers and hard iron

A magnetometer measures the Earth’s very weak magnetic field, so iron parts and currents on the drone disturb it easily. Hard iron is a constant field from permanent magnets or magnetised parts; it makes the data collected while rotating form a circle with a shifted centre. Soft iron from magnetically permeable materials stretches the circle into an ellipse. Compass calibration in ArduPilot or PX4 therefore asks you to rotate the drone in many directions, and ArduPilot has a CompassMot procedure to correct motor current effects.

import math

OFF, H = (12.0, -8.0), 35.0          # µT hard-iron offset and assumed horizontal field
samples = [(d, H * math.cos(math.radians(d)) + OFF[0], H * math.sin(math.radians(d)) + OFF[1])
           for d in range(0, 360, 10)]

def worst_error(off):
    errs = []
    for d, mx, my in samples:
        head = math.degrees(math.atan2(my - off[1], mx - off[0]))
        errs.append(abs((head - d + 180) % 360 - 180))
    return max(errs)

est = ((max(s[1] for s in samples) + min(s[1] for s in samples)) / 2,
       (max(s[2] for s in samples) + min(s[2] for s in samples)) / 2)
print(f"worst heading error without calibration: {worst_error((0, 0)):.1f} deg")
print(f"estimated offset ({est[0]:.1f}, {est[1]:.1f}) uT")
print(f"worst heading error after hard-iron correction: {worst_error(est):.1f} deg")

worst heading error without calibration: 24.3 deg
estimated offset (12.0, -8.0) uT
worst heading error after hard-iron correction: 0.0 deg

Compass heading is relative to magnetic north, which differs from true north by the magnetic declination, varying with place and time. ArduPilot has a built-in world magnetic model and can set it automatically. To check the value, use NOAA’s World Magnetic Model calculator (WMM2025).

Module lab

Common mistakes

Summary

• MEMS IMUs have bias, scale factor and noise errors.
• Six-position calibration finds bias and scale factor from ±g on every axis.
• Allan deviation separates angle random walk and bias instability from at-rest data.
• Hard iron shifts the centre of the compass data circle and can make heading badly wrong; calibrate and watch for interference.

Check your understanding

1. An axis reads +9.90 and −9.70 m/s². What is the bias?
2. From question 1, what is the scale factor (g = 9.80665)?
3. What noise type does the −1/2 slope region of an Allan deviation plot represent?
4. mx ranges from −20 to 40 µT. What is the hard-iron offset on this axis?
5. How do hard iron and soft iron change the shape of compass data differently?