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IMU Error Models and Preintegration ​

IMU Error Models and Preintegration curated visual

Visual: IMU preintegration timeline from high-rate samples through bias/noise model to preintegrated factor between keyframes, plus Allan variance cue.

An IMU is the short-term motion backbone for localization. It measures angular rate and specific force at high rate, but its errors integrate quickly into attitude, velocity, and position drift. Correct IMU modeling is the difference between a factor graph that bridges GNSS/LiDAR gaps and one that becomes confident in the wrong trajectory.

1. Measurement Model ​

For body frame B and world frame W:

omega_m = omega_true + b_g + n_g
a_m     = R_WB^T * (a_W - g_W) + b_a + n_a

where:

  • omega_m: measured angular velocity
  • a_m: measured specific force
  • b_g, b_a: gyroscope and accelerometer biases
  • n_g, n_a: white measurement noise
  • R_WB: rotation from body to world
  • g_W: gravity in world coordinates

The nominal strapdown dynamics are:

R_dot = R * skew(omega_m - b_g - n_g)
v_dot = R * (a_m - b_a - n_a) + g
p_dot = v

Biases are commonly modeled as random walks:

b_g_dot = n_bg
b_a_dot = n_ba

Sensor Model Impact ​

TaskWhy the model matters
PerceptionLiDAR deskew, radar Doppler compensation, and camera projection during motion rely on high-rate attitude and velocity.
SLAMIMU preintegration constrains motion between visual/LiDAR keyframes and improves initialization.
MappingMotion-compensated scans prevent bent walls and doubled structures in maps.
ValidationBias, vibration, saturation, and timestamp tests are required before trusting covariance during outages.

2. Error Sources ​

ErrorModelEffect
White noisen ~ N(0, sigma^2)random angle/velocity error that accumulates with integration
Biasadditive offsetgyro bias creates attitude drift; accel bias creates velocity and position drift
Bias random walkb_dot = n_bbias changes over time, temperature, vibration, and aging
Scale factorm = (I + S) trueacceleration or angular-rate magnitude errors
Axis misalignmentnon-orthogonal sensitivity axescross-axis coupling during turns and vibration
g sensitivitygyro affected by accelerationyaw/roll error under acceleration or vibration
Saturation/clippingmeasurement range exceededunrecoverable integration error
Quantizationfinite ADC/resolutionlow-amplitude noise floor
Timestamp errorsample time wrongphase lag and deskew error

A full calibration model can be written:

omega_m = M_g * omega_true + G_g * a_true + b_g + n_g
a_m     = M_a * R_WB^T*(a_W - g_W) + b_a + n_a

Many estimators only use additive bias and white noise online. Scale, misalignment, and g-sensitivity should still be calibrated offline or absorbed into conservative process noise.

3. Noise Density and Discrete Covariance ​

Datasheets often report noise density:

gyro_noise_density: rad/s/sqrt(Hz)
accel_noise_density: m/s^2/sqrt(Hz)

For samples at rate f_s with bandwidth near Nyquist, a rough discrete standard deviation is:

sigma_sample ~= noise_density * sqrt(f_s)

For continuous-time covariance used in propagation:

Q_c = diag(sigma_g^2, sigma_a^2, sigma_bg^2, sigma_ba^2)
Q_d ~= G * Q_c * G^T * dt

Conventions differ by library. Some APIs expect continuous noise densities, some expect per-sample covariance, and some expect covariance multiplied by dt. Verify with the library documentation and a stationary dataset.

4. Allan Variance Concepts ​

Allan deviation is a time-domain method for identifying inertial noise processes from long stationary logs. For cluster time tau, it measures how the average sensor output changes between adjacent time windows.

Log-log Allan deviation slopes indicate noise type:

SlopeCommon IMU interpretation
-1quantization noise
-1/2angle random walk or velocity random walk
0bias instability
+1/2rate random walk / acceleration random walk
+1rate ramp

Practical guidance:

  • Record hours of stationary data at the deployment sample rate.
  • Keep temperature stable or log temperature separately.
  • Estimate noise terms from the slope regions, then validate in motion.
  • Inflate stationary-derived values for vehicle vibration and thermal cycling, often by 2x to 20x depending on platform and mount.

Allan variance characterizes stochastic noise. It does not automatically capture deterministic vibration harmonics, poor mounting, clipping, time offset, or thermal transients.

5. Preintegration Motivation ​

IMUs run at 100 to 1000 Hz, while visual/LiDAR/GNSS keyframes may arrive at 1 to 30 Hz. Adding every IMU sample as a graph variable is inefficient. Preintegration summarizes many IMU samples between keyframes i and j into one relative motion factor.

IMU samples i...j
  -> preintegrated delta_R_ij, delta_v_ij, delta_p_ij
  -> one factor between states at i and j

The key idea from on-manifold preintegration is that the deltas are expressed relative to the starting body frame and can be corrected for small bias changes without reintegrating every sample.

6. Preintegrated Measurements ​

Using bias linearization point b_hat:

Delta_R_ij = product_k Exp((omega_k - b_g_hat) * dt)

Delta_v_ij = sum_k Delta_R_ik * (a_k - b_a_hat) * dt

Delta_p_ij = sum_k [Delta_v_ik * dt
                 + 0.5 * Delta_R_ik * (a_k - b_a_hat) * dt^2]

Bias correction:

delta_b = b_current - b_hat

Delta_R(b) ~= Delta_R(b_hat) * Exp(J_R_bg * delta_b_g)
Delta_v(b) ~= Delta_v(b_hat) + J_v_ba * delta_b_a + J_v_bg * delta_b_g
Delta_p(b) ~= Delta_p(b_hat) + J_p_ba * delta_b_a + J_p_bg * delta_b_g

The preintegrated covariance is propagated at IMU rate:

P_k+1 = A_k * P_k * A_k^T + B_k * Q_imu * B_k^T

where A_k and B_k are Jacobians of the discrete preintegration dynamics.

7. Factor Residual ​

For states:

x_i = (R_i, p_i, v_i, b_i)
x_j = (R_j, p_j, v_j, b_j)

The IMU factor residual compares predicted relative motion to preintegrated motion:

r_R = Log( Delta_R_ij(b_i)^T * R_i^T * R_j )

r_v = R_i^T * (v_j - v_i - g * dt_ij) - Delta_v_ij(b_i)

r_p = R_i^T * (p_j - p_i - v_i*dt_ij - 0.5*g*dt_ij^2)
      - Delta_p_ij(b_i)

Bias evolution is usually a separate between factor:

r_bg = b_g_j - b_g_i
r_ba = b_a_j - b_a_i

with covariance from the bias random-walk model.

8. Gravity Alignment and Observability ​

Accelerometers measure specific force, not gravity-free acceleration. At rest:

a_m ~= R_WB^T * (-g_W) + b_a

This lets the estimator align roll and pitch to gravity, but yaw remains unobservable from gravity alone. Yaw needs magnetometer, GNSS course, wheel motion, LiDAR/visual constraints, radar Doppler, or map matching.

Initialization risks:

  • Incorrect gravity sign or ENU/NED convention flips roll/pitch.
  • Stationary initialization cannot separate accelerometer bias from small tilt perfectly.
  • Constant straight motion provides weak yaw observability.
  • Visual-inertial monocular systems also need scale and gravity alignment.

For ground vehicles, level constraints can help but must account for slopes, ramps, aircraft stand drainage grades, and vehicle suspension motion.

9. Vibration, Mounting, and Filtering ​

Vehicle IMUs experience engine vibration, tire impacts, washboard surfaces, jet-blast vibration, and structural resonances. These are often colored and axis-dependent, not white Gaussian noise.

Operational checks:

  • Inspect PSD and spectrograms, not only Allan deviation.
  • Mount the IMU rigidly near the vehicle body reference when possible.
  • Avoid flexible brackets and long lever arms that amplify angular vibration.
  • Check for clipping during bumps, braking, and sharp turns.
  • Ensure anti-alias filtering matches sample rate.

If vibration is severe, increasing white noise alone may not fix the estimator. Consider mechanical isolation, notch/low-pass filters with known delay, better mounting, or modeling colored noise.

10. Practical GTSAM-Style Noise Guidance ​

Typical setup:

accelerometerCovariance = sigma_a^2 * I
gyroscopeCovariance     = sigma_g^2 * I
integrationCovariance   = small numerical integration noise
biasAccCovariance       = sigma_ba^2 * I
biasOmegaCovariance     = sigma_bg^2 * I

Guidance:

  • Use datasheet noise density as a starting point.
  • Confirm units: deg/hr, deg/s/sqrt(Hz), rad/s/sqrt(Hz), mg, and m/s^2 are easy to mix.
  • Estimate stationary noise and bias stability with Allan deviation.
  • Validate in vehicle logs by checking residual consistency and bias estimates.
  • Bias process noise too small prevents adaptation after temperature/vibration changes.
  • Bias process noise too large lets bias absorb real motion and weakens inertial constraints.

Health monitors:

|b_g| and |b_a| magnitude and derivative
IMU factor residual NIS
gravity norm residual during stationary periods
sample interval jitter and dropped samples
saturation counters
temperature versus bias

11. Failure Modes ​

Failure modeCauseMitigation
Fast yaw driftgyro z bias or yaw unobservableLiDAR/vision/GNSS/radar heading constraints, bias tuning
Position blows up in outageaccelerometer bias and gravity misalignmentrobust initialization, external updates, conservative covariance
Deskew bends scanstime offset or wrong IMU-LiDAR extrinsictemporal calibration and projection replay
Optimizer divergesbad initial gravity/bias or underestimated noiseinitialize stationary, inflate noise, reject bad intervals
Bias estimate jumpsvibration, clipping, over-loose bias random walkmount/filters, saturation checks, better process model
Roll/pitch sign wrongENU/NED or frame convention errorexplicit gravity tests and frame unit tests
Preintegration covariance wrongcontinuous/discrete unit mixstationary and motion residual validation

12. Sources ​

Public research notes collected from public sources.

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