Why Inertial Navigation Drifts

Inertial navigation is powerful because it is self-contained over short periods. It also drifts because tiny sensor errors are integrated. Woodman’s Cambridge report demonstrates how MEMS-based strapdown INS errors can grow very quickly, including an example where average position error grows to over 150 m after 60 seconds for a simple INS based on an Xsens Mtx IMU.¹

The problem

Double integration is unforgiving. If acceleration has a constant bias, position error grows roughly with $t^2$. If gyro bias causes attitude error, gravity can be projected into the wrong axis and create an apparent acceleration.

Accelerometer bias

If acceleration bias is constant:

$$\tilde{a}=a+b_a$$

Velocity error grows as:

$$\delta v(t)=b_a t$$

Position error grows as:

$$\delta p(t)=\frac{1}{2}b_a t^2$$

A bias of only $0.01\,m/s^2$ gives:

$$\delta p(60)=\frac{1}{2}(0.01)(60)^2=18\,m$$

Gyro bias and gravity leakage

A gyro bias creates attitude error:

$$\delta \theta(t) \approx b_g t$$

A small tilt error leaks gravity into the horizontal acceleration estimate:

$$\delta a \approx g\delta\theta$$

So the resulting position error can grow very rapidly. This is why orientation errors are so damaging in strapdown INS.¹

Interactive demo: stationary IMU drift simulator

The simulator below keeps the true object fixed at the origin and lets only the inertial estimate move.

What to watch

• With acceleration bias only, position error grows in the same rough order as the $t^2$ equation above.
• With gyro bias, attitude error leaks gravity into the wrong axis and drift becomes worse.
• Higher sample rate alone does not remove bias.
• ZUPT and external correction limit the stationary drift.

Try this

Set acceleration bias to $0.01\,m/s^2$ and run for 60 seconds. The final error should be in the same order as the equation above. Then add gyro bias and compare how much worse attitude-induced drift can be.

Where this breaks

Real inertial errors include bias instability, scale-factor error, misalignment, vibration, quantisation and temperature effects. Analog Devices’ sensor-error summary is a useful practical reference for the kinds of errors that appear in real sensors.²

Footnotes

1. Oliver J. Woodman, “An introduction to inertial navigation”, University of Cambridge Computer Laboratory Technical Report UCAM-CL-TR-696, 2007. https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-696.html - Accessible report on MEMS strapdown INS error characteristics and drift growth. Accessed 2026-06-11. ↩ ↩²

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. ↩