M1-T8: IMU, barometer, magnetometer and GNSS

Learning objectives

After this lesson, you should be able to:

• State what each of the IMU, barometer, magnetometer and GNSS measures and in which frame.
• Name the dominant error of each and the environment that triggers it.
• Convert pressure to altitude and compute a tilt-compensated heading.
• Explain which state each sensor makes observable and which it does not.

Conceptual explanation

The IMU measures angular rate and specific force at high rate in the body frame. The barometer measures static pressure, which maps to altitude. The magnetometer measures the magnetic field vector in the body frame, which gives heading when the field is known. GNSS measures position and velocity in a global frame.

Engineering intuition

Think of two kinds of information. Relative sensors (IMU) are smooth and fast but drift; absolute sensors (GNSS, barometer, magnetometer) are noisy or slow but do not drift. A fusion filter uses the absolute ones to pin down what the relative one cannot hold, and the relative one to fill the gaps between absolute updates. The failure modes are complementary on purpose: an IMU never loses signal, GNSS never drifts.

The trap is that each absolute sensor is absolute about something specific: the barometer about pressure altitude (not height above ground), the magnetometer about the local field direction (not true north without declination), GNSS about the antenna position (not the centre of mass). Offsets and reference frames are where integration bugs live.

Mathematical formulation

Pressure to altitude

For the International Standard Atmosphere below 11 km, with sea-level pressure $p_{0}=101325$ Pa:

A 1 hPa weather change moves the reading by roughly 8 m, so barometric altitude is a relative measurement over a flight, anchored at take-off.

Tilt-compensated heading

With roll $\phi$, pitch $\theta$ and body-frame magnetometer $m=(m_{x},\,m_{y},\,m_{z})$ in FRD, rotate the field into the horizontal plane then take the angle:

GNSS accuracy

Position error ≈ DOP × range error (lesson M2-T1); velocity from Doppler is typically far more accurate than position differences.

$p_{0}$

reference sea-level pressure (Pa); actual local value differs by weather.

$\phi$, $\theta$

roll and pitch from the attitude estimate: a heading error appears if they are wrong.

Architecture and data flow

In PX4, drivers publish each sensor to uORB; EKF2 fuses them: IMU for propagation, barometer for height, magnetometer for yaw, GNSS for horizontal position and velocity (and optionally height), with innovation gating on each (lesson M6-T5). When you remove GNSS (lesson M2-T6) or distrust the magnetometer, other sources (vision, LiDAR, external yaw) must supply what they provided.

Real-world UAV applications

• Understanding which EKF2 fusion switches to change for GNSS-denied flight.
• Diagnosing altitude jumps from propwash on the barometer.
• Calibrating the magnetometer away from the airframe wiring.
• Choosing between barometer and rangefinder for height near the ground.

Implementation example

This runs as shown: ISA pressure-to-altitude, its sensitivity, and a tilt-compensated heading checked on a synthetic field (northern-hemisphere style field with inclination).

import numpy as np

def pressure_altitude(p_pa, p0=101325.0): return 44330.0 * (1.0 - (p_pa / p0) ** (1 / 5.255))
assert abs(pressure_altitude(101325.0)) < 1e-9
h = pressure_altitude(89875.0); assert 1000 < h < 1100                     # about 1 km: ISA pressure at 1000 m is 89875 Pa
dh = pressure_altitude(100325.0) - pressure_altitude(101325.0); assert 80 < dh < 90      # 10 hPa lower pressure is about 83 m higher (about 8.3 m per hPa): weather matters

def rot_frd(roll, pitch, yaw):     # body (FRD) -> NED, ZYX
    cr, sr, cp, sp, cy, sy = np.cos(roll), np.sin(roll), np.cos(pitch), np.sin(pitch), np.cos(yaw), np.sin(yaw)
    return np.array([[cy*cp, cy*sp*sr - sy*cr, cy*sp*cr + sy*sr], [sy*cp, sy*sp*sr + cy*cr, sy*sp*cr - cy*sr], [-sp, cp*sr, cp*cr]])

def heading(m, roll, pitch):
    mx, my, mz = m
    mhx = mx * np.cos(pitch) + my * np.sin(pitch) * np.sin(roll) + mz * np.sin(pitch) * np.cos(roll)
    mhy = my * np.cos(roll) - mz * np.sin(roll)
    return np.arctan2(-mhy, mhx)

B_ned = np.array([20.0, 0.0, 45.0])                                        # microtesla: north, east, down (inclination about 66 degrees)
for yaw_deg in (0, 45, 90, 200, -120):
    roll, pitch, yaw = np.radians([12, -8, yaw_deg])
    m_body = rot_frd(roll, pitch, yaw).T @ B_ned                            # the field as the body-fixed sensor sees it
    est = np.degrees(heading(m_body, roll, pitch)); err = (est - yaw_deg + 180) % 360 - 180
    assert abs(err) < 1e-9, (yaw_deg, est)
print("tilt-compensated heading recovers yaw for tilted attitudes")
m_body = rot_frd(*np.radians([12, -8, 45])).T @ B_ned
assert abs(np.degrees(heading(m_body, 0.0, 0.0)) - 45) > 3                  # ignoring tilt gives a visible heading error

Common engineering mistakes

Failure modes

Debugging approach

1. Plot each sensor against the EKF estimate and against another sensor of the same quantity.
2. Run the magnetometer-versus-throttle test.
3. Inspect IMU spectra and clipping counters.
4. Log GNSS accuracy estimates and satellite counts next to position error.

Practical exercises

1. Convert 85000 Pa to altitude with the ISA formula and compare with 1 hPa steps.
2. Show numerically how a $10^{\circ}$ tilt error causes a heading error with an inclination of $66^{\circ}$.
3. List which state each of the four sensors makes observable and which remain unobservable without another sensor.
4. Estimate how fast barometer-only altitude can drift in a 10 hPa per hour weather change.

Interview questions

Write an answer in your own words, then reveal the model answer. Nothing is sent anywhere.

References and further reading

Primary papers, official documentation and standards. Each claim above that is specific to a tool or version points here.

• PX4: Using the EKF2 estimator (sensor fusion options)
• PX4: compass configuration and calibration
• $P$. $D$. Groves, Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd ed., Artech House, 2013, chapters 4 and 6.
• NOAA: magnetic field calculators (declination and inclination)

What’s next