Attitude and heading reference systems

"I knew exactly how the drone was tilted and had no idea which way it faced. Gravity is a generous witness; it will tell you everything except where north went."

A Well-Oriented but Lost AI Agent

From a filter to a system

Section 24.2 gave you estimators that fuse a gyroscope with an accelerometer to track orientation over time. An AHRS is the engineering wrapper around such an estimator that makes it deployable. Concretely it adds four things the bare filter lacks. First, an initialization (often called alignment): at power-on there is no history, so the system must derive a starting orientation from a single static snapshot of the sensors. Second, a third reference vector, the magnetometer, because two of the three orientation axes are otherwise unobservable (the next subsection is entirely about why). Third, calibration and disturbance handling, since a magnetometer near a motor or a laptop reports a field that has nothing to do with north. Fourth, degraded modes and health flags, so that when the magnetic environment is untrustworthy the system tells you it is coasting rather than pretending. A commercial AHRS module such as the VectorNav VN-100 or the Xsens MTi packages exactly these behaviors behind a serial stream of quaternions; the point of this section is to understand what that stream is quietly protecting you from.

Why heading is the hard half

Attitude and heading feel like one problem, but they have different amounts of evidence behind them. Consider what each sensor observes. The gyroscope measures angular rate on all three axes, so integrating it moves the estimate correctly in the short term but accrues drift on all three (Section 24.2). The accelerometer, at rest, measures the gravity direction, a fixed vector in the navigation frame. Aligning the estimated gravity to the measured gravity pins down two rotational degrees of freedom: roll and pitch. It cannot pin the third. Rotating the device about the gravity axis (a change of heading) leaves the gravity vector pointing in exactly the same place, so the accelerometer sees no difference at all. Yaw is invisible to gravity.

This is an observability statement, and it is the crux of the whole section. With only a gyroscope and an accelerometer, roll and pitch are corrected by an absolute reference and stay bounded, while heading has no absolute reference and drifts without limit, at the gyroscope's bias rate, forever. A cheap MEMS gyro drifting a degree or two per minute will have your heading a full quadrant wrong within the hour, even though your attitude stays crisp. The only way to make heading observable is to add a second, non-parallel reference vector fixed in the world. The Earth's magnetic field is the one every device can carry: a vector that points roughly toward magnetic north with a downward tilt, and crucially one that is not parallel to gravity, so it supplies the missing degree of freedom.

Making the magnetometer tell the truth

Two reference vectors give you a full orientation through the classic solution to Wahba's problem: find the single rotation that best aligns the measured gravity-and-field pair to their known navigation-frame directions (the TRIAD and QUEST algorithms do this in closed form, and an AHRS uses one at startup for instant alignment). But this only works if the magnetometer is honest, and out of the box it is not. Three corruptions must be handled before the field vector means anything.

Hard-iron distortion is a constant offset added by permanently magnetized material soldered next to the sensor (speaker magnets, steel screws). It shifts the whole field measurement, so the locus of readings taken while rotating the device is a sphere not centered at the origin. Soft-iron distortion comes from nearby ferromagnetic material that warps the field, stretching that sphere into an ellipsoid. Both are fixed relative to the device and are removed by a one-time calibration: rotate the unit through all orientations, fit an ellipsoid to the cloud of readings, and store the offset and shape correction that map it back to a unit sphere. The third correction is magnetic declination: the field points at magnetic north, which differs from true (geographic) north by a location-dependent angle, tens of degrees near the poles. You add the declination for your location, obtained from a model like the World Magnetic Model, to turn magnetic heading into true heading before handing it to a map.

Even calibrated, the field must be de-rotated into the horizontal plane before it yields a heading, because a magnetometer tilted with the device reads a field pointing partly down. Listing 24.1 performs this tilt compensation: it recovers roll and pitch from gravity, uses them to level the magnetometer, and only then computes heading. This is the computation an AHRS runs every cycle to feed its yaw correction.

import numpy as np

def tilt_compensated_heading(acc, mag, declination_deg=0.0):
    """True heading in degrees from an accelerometer and a (calibrated)
    magnetometer. acc and mag are 3-vectors in the sensor frame."""
    ax, ay, az = acc / np.linalg.norm(acc)          # gravity direction
    roll  = np.arctan2(ay, az)                       # tilt about forward axis
    pitch = np.arctan2(-ax, np.hypot(ay, az))        # tilt about right axis

    cr, sr = np.cos(roll),  np.sin(roll)
    cp, sp = np.cos(pitch), np.sin(pitch)
    mx, my, mz = mag
    mh_x = mx*cp + my*sr*sp + mz*cr*sp               # level the field into the
    mh_y = my*cr - mz*sr                             # local horizontal plane
    heading = np.degrees(np.arctan2(-mh_y, mh_x))
    return (heading + declination_deg) % 360.0

# Device pitched 30 deg forward, magnetometer reads a north-ish tilted field
acc = np.array([-0.5, 0.0, 0.866]) * 9.81
mag = np.array([18.0, -4.0, 42.0])                   # microtesla, sensor frame
print(round(tilt_compensated_heading(acc, mag, declination_deg=11.0), 1))

import imufusion, numpy as np
ahrs = imufusion.Ahrs()                       # gyro + accel + magnetometer
for gyro, accel, mag in stream:               # deg/s, g, microtesla
    ahrs.update(gyro, accel, mag, dt=1/100)
    q = ahrs.quaternion.wxyz                   # calibrated orientation
    euler = ahrs.quaternion.to_euler()         # roll, pitch, heading

Rejection, degraded modes, and staying honest

The reason a serious AHRS is more than tilt-compensated heading plus a filter is that indoor and vehicular magnetic environments are hostile. A steel doorway, an elevator, or a running motor can bend the local field by tens of microtesla, swamping the Earth's roughly 50-microtesla signal. If the filter naively trusts the magnetometer, heading snaps toward the disturbance and then unwinds slowly, a signature wobble that ruins a trajectory. The defense is disturbance rejection: on every sample the AHRS checks whether the measured field magnitude and inclination match the expected local values, and if they do not, it temporarily ignores the magnetometer and lets the gyroscope carry heading on its own. Because gyro heading drifts only slowly, coasting for a few seconds through a bad patch is far better than believing it. This is a direct application of gating an outlier measurement, the same robustness idea that appears in the Kalman family of Chapter 9. A well-designed AHRS also exposes a health flag when it is coasting, so downstream fusion (for example with GNSS in Chapter 25) can down-weight heading rather than trust a number the AHRS itself does not.

What's Next

An AHRS solves orientation: which way the device is tilted and which way it faces. It says nothing about where the device is. In Section 24.4, we attempt the far harder step of integrating acceleration into velocity and position, and meet the unforgiving reason pure inertial navigation drifts away in seconds: the same gravity that anchored our attitude now contaminates every position estimate through the smallest orientation error.