IMU Calibration: Bias, Scale and Misalignment

A cheap IMU is not useless, but it must be treated honestly. Bias, scale-factor error, misalignment, temperature drift and noise all affect navigation. Analog Devices describes null bias, scale-factor error and noise as major inertial-sensor error sources.¹ Their gyro calibration note gives practical equations for null output, scale factor and temperature compensation.²

The problem

When an accelerometer is stationary in many orientations, the magnitude of the true specific force should be approximately $g$. But raw measurements often form an offset, stretched ellipsoid rather than a clean sphere.

The model

A common accelerometer calibration model is:

$$\mathbf{y}=\mathbf{M}\mathbf{a}+\mathbf{b}+\boldsymbol{\eta}$$

where:

• $\mathbf{y}$ is raw measurement.
• $\mathbf{a}$ is true specific force.
• $\mathbf{M}$ captures scale and misalignment.
• $\mathbf{b}$ is bias.
• $\boldsymbol{\eta}$ is noise.

Corrected acceleration can be estimated by:

$$\hat{\mathbf{a}}=\mathbf{M}^{-1}(\mathbf{y}-\mathbf{b})$$

A six-position accelerometer calibration idea uses orientations where each axis is aligned with $+g$ and $-g$. A more visual method fits an ellipsoid to stationary data and transforms it toward a sphere.

For a gyro with voltage output, the calibration note gives the basic form:

$$\omega = \frac{V_{gyro}-V_0}{S}$$

where $V_0$ is null output and $S$ is scale factor.²

Interactive demo: accelerometer ellipsoid calibration

The demo below uses three 2D projections so the calibration effect remains visible without a heavy 3D plotting stack.

What to watch

• Raw points shift and stretch as bias and scale error increase.
• Pressing calibrate recentres the cloud and reduces magnitude error.
• Simple per-axis calibration helps a lot for bias and scale, but misalignment leaves residual structure.
• The before/after RMS readout makes the improvement measurable rather than purely visual.

Try this

Add a large X-axis bias. The raw cloud should shift away from the origin. Press calibrate and watch the corrected cloud centre itself. Then add misalignment and note that simple per-axis calibration is no longer enough.

Where this breaks

Temperature-dependent bias, dynamic acceleration, vibration and imperfect orientation data make real calibration harder. This post should point readers toward the next level rather than oversell a simple six-point fit.

Footnotes

1. 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. ↩

2. Analog Devices, “AN-1049: Calibrating iMEMS Gyroscopes”. https://www.analog.com/en/resources/app-notes/an-1049.html - Gyro null output, scale factor and temperature compensation reference. Accessed 2026-06-11. ↩ ↩²