Accelerometers, gyroscopes, magnetometers

"I know precisely how fast I am spinning. I am simply no longer certain which way is down, where north lies, or whether I have moved at all."

An Inertially Confident AI Agent

Three sensors, three physical quantities

The triad is usually sold as a "9-axis IMU" (three axes each), and treating it as one device hides the physics that determines how you must process it. Each sensor answers a different question about the same rigid body, and each is silent about what the others report.

The accelerometer measures specific force: the non-gravitational force per unit mass acting on the sensor case, expressed in the sensor's own axes. This is the point every newcomer trips on. A phone lying flat and perfectly still on a table reads roughly \(9.81\ \mathrm{m/s^2}\) upward, not zero, because the table pushes up on it while gravity is not a force the device can feel. In free fall the same device reads zero. The clean statement is \[ \tilde{\mathbf a} = \mathbf a_{\text{lin}} - \mathbf g + \mathbf b_a + \mathbf n_a, \] where \(\mathbf a_{\text{lin}}\) is true linear acceleration, \(\mathbf g\) is the local gravity vector, \(\mathbf b_a\) is a slowly varying bias, and \(\mathbf n_a\) is white noise. Because gravity is always in the reading, an accelerometer at rest is a superb tilt sensor: the direction of the measured vector points along "down." Under motion, gravity and linear acceleration are tangled, and untangling them is the whole job of Section 23.3.

The gyroscope measures angular rate \(\boldsymbol\omega\), the rotational velocity of the case about its three axes, in radians or degrees per second: \[ \tilde{\boldsymbol\omega} = \boldsymbol\omega + \mathbf b_g(t) + \mathbf n_g. \] It knows nothing about position or absolute orientation. To get an angle you must integrate rate over time, and integration is where a gyroscope's small, drifting bias \(\mathbf b_g(t)\) becomes ruinous: a constant \(0.01^\circ/\mathrm{s}\) bias, tiny on any single sample, accumulates to \(36^\circ\) of heading error in one hour. That single fact drives almost every design choice in Chapter 24.

The magnetometer measures the local magnetic flux density \(\mathbf m\), a vector combining the Earth's field (about 25 to 65 microtesla, tilted by the local inclination angle) with fields from nearby motors, speakers, steel beams, and the device's own currents: \[ \tilde{\mathbf m} = \mathbf m_{\text{earth}} + \mathbf m_{\text{disturb}} + \mathbf b_h + \mathbf n_h. \] On a clean day the horizontal component points toward magnetic north, which is why the magnetometer is the only member of the triad with an absolute, non-drifting heading reference. It pays for that with vulnerability to disturbance: walk past a refrigerator and your "north" swings.

Inside the MEMS: where the numbers come from

Understanding the transduction explains the error terms rather than leaving them as mysterious datasheet lines. Consumer accelerometers and gyroscopes are MEMS devices, microscopic mechanical structures etched into silicon, and their imperfections follow directly from being tiny springs and masses.

A MEMS accelerometer is a proof mass on silicon springs forming a differential capacitor. Specific force displaces the mass, changing the capacitance, which readout electronics convert to a number. Because the spring stiffness drifts with temperature and the mass is measured in femtofarads, the zero point wanders: that is your bias, and its temperature dependence is why datasheets quote bias over a temperature range. A MEMS gyroscope exploits the Coriolis effect: a mass is driven into continuous vibration, and when the case rotates, the Coriolis force \(\mathbf F_c = -2m\,\boldsymbol\omega \times \mathbf v\) pushes the mass sideways in proportion to angular rate. The sideways deflection is again read capacitively. Coriolis forces are minuscule, so gyroscopes are inherently noisier and more bias-prone than accelerometers, and they are sensitive to vibration that mimics the drive motion. Magnetometers use a different physics entirely, usually the Hall effect or anisotropic magnetoresistance, where a magnetic field bends charge carriers or changes a film's resistance; they are not mechanical, so they do not share the MEMS vibration sensitivity, but they saturate near strong fields.

The measurement model you will actually code against

For processing, collapse each sensor to bias plus white noise (scale and axis misalignment are real too, but they belong to Section 23.4). Two numbers characterize the random part, and both come straight from the datasheet. Noise density \(\rho\) is the white-noise level per root hertz of bandwidth: an accelerometer at \(150\ \mu g/\sqrt{\mathrm{Hz}}\) has a per-sample standard deviation of \(\sigma = \rho\sqrt{\mathrm{BW}}\), so wider bandwidth buys more noise. This is why you never sample an IMU faster than you need and then filter to the band that carries your signal. Bias instability is the floor of the Allan-deviation curve: the slowest, irreducible wander of the zero point, the term that no averaging removes and that dead reckoning integrates into unbounded error. The bias itself is well modeled as a random walk, \(\mathbf b_{k+1} = \mathbf b_k + \mathbf w_k\), which is exactly the process-noise structure a Kalman filter in Chapter 9 is built to track.

The code below takes a recording from a stationary IMU and recovers these constants: the mean is the bias, the standard deviation over a fixed bandwidth gives the noise density, and watching how the average of ever-longer windows behaves reveals the bias-instability floor. Run it on your own phone logs and the numbers should land within a factor of two of the datasheet.

import numpy as np

def characterize_axis(x, fs):
    """Estimate bias, white-noise density, and a crude bias-instability
    floor from a stationary single-axis IMU recording x sampled at fs Hz."""
    x = np.asarray(x, dtype=float)
    bias = x.mean()                                  # zero-point offset
    resid = x - bias
    sigma = resid.std()                              # per-sample white noise
    noise_density = sigma / np.sqrt(fs / 2.0)        # units per sqrt(Hz)

    # Overlapping-average stability: std of block means vs block length.
    # The curve stops falling near the bias-instability floor.
    floors = []
    for m in (10, 50, 200, 1000):
        if len(resid) < 2 * m:
            break
        k = len(resid) // m
        block_means = resid[:k * m].reshape(k, m).mean(axis=1)
        floors.append((m / fs, block_means.std()))
    return bias, noise_density, floors

fs = 200.0                                           # Hz
gyro_z = np.random.normal(0.02, 0.05, size=60 * int(fs))  # 60 s at rest
bias, nd, floors = characterize_axis(gyro_z, fs)
print(f"bias        = {bias:.4f} deg/s")
print(f"noise dens. = {nd*1000:.2f} mdps/sqrt(Hz)")
for tau, dev in floors:
    print(f"  tau={tau:5.2f}s  stability={dev*1000:.2f} mdps")

Why one sensor is never enough

Put the error signatures side by side and the case for the triad is immediate. Ask the accelerometer alone for orientation and it answers perfectly at rest but lies the moment the device accelerates, because it cannot tell a push from a change in "down." Ask the gyroscope alone and it tracks fast rotations beautifully for a few seconds, then its integrated bias slowly rotates your whole world away from truth. Ask the magnetometer alone for heading and it holds an absolute reference indefinitely, until someone opens a laptop nearby. No single stream is trustworthy across both time scales and all motions; each is reliable exactly where another is weak. That structural complementarity, not marketing, is why phones, drones, watches, and cars all carry the same nine axes, and why the downstream models of Chapter 26 are trained on all three at once rather than on any one alone.

What's Next

In Section 23.2, we make the "in the sensor's own axes" caveat precise. Every quantity in this section, specific force, angular rate, magnetic field, was reported in the moving body frame, and to reason about the world we must relate that frame to a fixed one. Coordinate frames and orientation representations are the machinery that turns three axis-locked streams into a statement about how a device is posed in space.