What an IMU Really Measures

An IMU usually contains accelerometers and gyroscopes. A gyroscope measures angular rate. An accelerometer does not directly give “change of velocity in the navigation frame”; it measures specific force in its own body frame. Groves’ inertial-sensor tutorial covers accelerometer and gyroscope technology, strapdown inertial navigation and attitude determination.¹

The problem

A stationary phone on a desk often reports an acceleration magnitude near $9.81\,m/s^2$. That can confuse beginners: if the phone is not moving, why is the accelerometer reading not zero?

The answer is that an accelerometer senses non-gravitational specific force. In a stationary body frame, gravity appears in the accelerometer measurement because the table is pushing up on the phone.

The model

A simplified accelerometer measurement model is:

$$\mathbf{f}_b = \mathbf{C}_{n}^{b}(\mathbf{a}_n - \mathbf{g}_n) + \mathbf{b}_a + \boldsymbol{\eta}_a$$

where:

• $\mathbf{f}_b$ is measured specific force in body frame.
• $\mathbf{C}_{n}^{b}$ rotates navigation-frame vectors into the body frame.
• $\mathbf{a}_n$ is acceleration in navigation frame.
• $\mathbf{g}_n$ is gravity.
• $\mathbf{b}_a$ is accelerometer bias.
• $\boldsymbol{\eta}_a$ is noise.

A gyro measurement model is:

$$\boldsymbol{\omega}_{meas}=\boldsymbol{\omega}_{true}+\mathbf{b}_g+\boldsymbol{\eta}_g$$

Analog Devices lists bias, scale-factor error and noise as major inertial-sensor error sources.²

Interactive demo: IMU specific-force cube

The embedded cube below keeps that sign convention explicit. Rotate the body frame, then compare the gravity vector in navigation coordinates against the specific-force readout reported in the body frame.

What to watch

• Rotating the cube changes which body axis carries the gravity-related reading.
• The magnitude stays close to $g$ when the device is stationary.
• Adding linear acceleration moves the vector away from pure gravity.
• Bias and noise create persistent offsets and jitter in the sensor readout.

Try this

Place the cube flat, then roll it $90^\circ$. The gravity-related reading should move from one axis to another. This is the intuition behind accelerometer-based tilt estimation.

Where this breaks

An accelerometer cannot distinguish gravity from acceleration by itself. A phone accelerating sideways and a tilted phone can produce similar measurements. This is why navigation systems need gyros, models, constraints and external measurements.

Footnotes

1. Paul D. Groves, “Navigation Using Inertial Sensors”, IEEE Aerospace and Electronic Systems Magazine, 2015. https://ieee-aess.org/media/navigation-using-inertial-sensors - Tutorial covering inertial sensor technology, strapdown navigation, alignment, zero updates, motion constraints, pedestrian dead reckoning and fault detection. Accessed 2026-06-11. ↩

2. Analog Devices, “What are the major error sources for inertial sensors?”. https://www.analog.com/en/resources/faqs/faq_what_are_the_major_error_sources_for_inertial.html - Inertial sensor bias, scale factor and noise reference. Accessed 2026-06-11. ↩