M2-T8: Dead reckoning and inertial drift

Learning objectives

After this lesson, you should be able to:

• Derive position error growth from a constant accelerometer bias, a gyro bias and white accelerometer noise.
• Estimate the time until inertial position error exceeds a given bound for a given IMU quality.
• Explain why attitude (gyro) errors eventually dominate.
• Fit a bias to a stationary log and predict drift.

Conceptual explanation

Dead reckoning is estimating position by integrating motion from a known starting point. For an IMU, errors in the measurement are integrated along with the signal. Three sources matter: accelerometer bias (a constant offset), accelerometer noise (random jitter) and gyroscope bias (a constant attitude-rate offset that corrupts the gravity projection).

Engineering intuition

The exponent counts integrations. A bias in acceleration is integrated twice: error ∝ $t^{2}$. White noise in acceleration integrates to a random walk in velocity (error ∝ $t^{1/2}$) and then to position $(\propto t^{3/2})$. A gyro bias is integrated once into tilt, which leaks gravity into horizontal acceleration, then twice more: three integrations, $t^{3}$. A cheap IMU can therefore look fine for 5 seconds and be metres off at 30.

Mathematical formulation

Accelerometer bias

Accelerometer white noise

For a noise density $N$ (m/s²/√Hz), velocity error has standard deviation $N\sqrt{t}$ and position error standard deviation:

Gyro bias

A constant gyro bias $b_{g}$ gives a tilt error $\delta \theta (t)=b_{g}\,t$, a spurious horizontal acceleration $g\,\delta \theta$, and:

$b_{a}$

accelerometer bias $(m/s^{2})$. 1 mg = $0.00981\,\mathrm{m}/\mathrm{s}^{2}$.

$b_{g}$

gyro bias (rad/s). 1°/s = 0.01745 rad/s.

$N$

accelerometer noise density (m/s²/√Hz): the “velocity random walk” in m/s/√Hz.

$g$

$9.81\,\mathrm{m}/\mathrm{s}^{2}$.

The cubic term dominates for $t$ above the crossover where ⅙ $g\,b_{g}\,t^{3}$ = ½ $b_{a}\,t^{2}$, that is $t=3\,b_{a}/(g\,b_{g})$. With $b_{a}=5$ mg and $b_{g}=0.1\,\mathrm{°}/\mathrm{s}$ this is about 86 s. Important: these formulas apply to residual bias after calibration and filtering; consumer MEMS devices can be a lot worse before calibration.

Architecture and data flow

Bias is not constant: it drifts (bias random walk) and depends on temperature. A good estimator therefore estimates the biases online while aiding is available (lesson M3-T7), then dead-reckons with the last estimate during an outage. The residual bias error at the moment aiding is lost sets the drift rate for the whole outage. That is why a short, noisy aiding window just before GNSS loss can be worse than a long clean one earlier.

Real-world UAV applications

• Specifying IMU grade from the maximum allowed outage: how long must the vehicle survive without aiding?
• Choosing when to trigger a failsafe: error bound versus time since last aiding.
• Judging whether a log’s drift is explained by bias or by something else (misalignment, timing).
• Deciding the update rate and covariance inflation in a GNSS-denied estimator.

Implementation example

This runs as shown: it simulates a stationary 1-D accelerometer with bias and noise, fits the bias from the first 10 seconds, and predicts the drift at 60 seconds from the formulas.

import numpy as np
rng = np.random.default_rng(2)
dt, T = 0.01, 60.0
b_a = 0.049                    # 5 mg bias (m/s^2)
N = 0.002                      # noise density, m/s^2/sqrt(Hz)
n = int(T / dt)
acc = b_a + N / np.sqrt(dt) * rng.standard_normal(n)      # stationary: true acceleration is 0

v = np.cumsum(acc) * dt
p = np.cumsum(v) * dt
print(f"position error at 60 s: {p[-1]:.1f} m")

# 1) fit the bias from the first 10 s of stationary data and predict the drift
b_hat = acc[: int(10 / dt)].mean()
pred = 0.5 * b_hat * T**2
print(f"bias estimate {b_hat/9.81*1000:.2f} mg, predicted {pred:.1f} m")
assert abs(pred - p[-1]) / p[-1] < 0.05                    # the bias model explains the drift

# 2) after removing the estimated bias, only noise remains: much smaller growth
acc2 = acc - b_hat
p2 = np.cumsum(np.cumsum(acc2) * dt) * dt
sigma_pred = N * T**1.5 / np.sqrt(3)
print(f"after bias removal: error {abs(p2[-1]):.3f} m, noise-only 1-sigma prediction {sigma_pred:.3f} m")
assert abs(p2[-1]) < 5 * sigma_pred + 0.5 and abs(p2[-1]) < 0.1 * p[-1]

Common engineering mistakes

Failure modes

Debugging approach

1. Place the vehicle still, log for a minute and compute the mean and Allan deviation of each axis.
2. Fit ½ $b\,t^{2}$ to the integrated position of the stationary data to recover $b_{a}$.
3. Check the temperature trace against bias in a warm-up test.
4. Verify that the estimator’s covariance during a simulated outage grows at least as fast as the true error.

Practical exercises

1. How long until position error exceeds 10 m for a 2 mg bias? And 1 m?
2. Find the crossover time at which gyro-bias drift overtakes accelerometer-bias drift for $b_{a}=2$ mg and $b_{g}=0.05\,\mathrm{°}/\mathrm{s}$.
3. If you must survive a 30 s GNSS outage within 5 m, what accelerometer bias residual is acceptable?
4. Use the Sensor Fusion Simulator to reproduce the $t^{2}$ growth with fusion disabled.

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.

• $P$. $D$. Groves, Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd ed., Artech House, 2013.
• IEEE Std 952-1997, Guide for the specification of inertial sensors (Allan variance method; check current revision)
• $T$. $D$. Barfoot, State Estimation for Robotics, Cambridge University Press, 2017.

What’s next