Logmap IMU Preintegration Results

This report summarizes the executed Galilean IMU factor NEES notebook. All four GTSAM preintegration backends use the default $SE_2(3)$ Logmap factor error,

error = state_j.logmap(predictedState_j);

The notebook is the executable source of truth. Its mechanical Monte Carlo, statistics, tables, and Plotly code are kept in galilean_imu_factor_nees.py.

The Galilean row is the companion paper’s $\mathrm{Gal}(3)\times\mathbb R^6$ direct-product model. It retains Delama et al.'s held-input composition and uses Brossard et al.'s endpoint and rotating-frame structure, but its physical bias model, left-invariant covariance, and right correction are the companion paper’s formulation.

How to read the results¶

Normalized Estimation Error Squared (NEES) tests whether a residual and its predicted covariance agree. The expected mean is the residual dimension: 9 for the ordinary IMU factor and 15 for the Combined factor. A value nearest that mean is best; a smaller NEES is not automatically better.

Physical position and velocity RMS errors are reported separately. A residual chart cannot repair deterministic integration error.

One-second inertial stress test¶

This experiment integrates simultaneous high rotation and acceleration at 20 Hz. The expected 95% interval for mean 9D NEES is [8.849, 9.152].

The established backends are overconfident in this deliberately stressful finite-rate case. Galilean preintegration removes most of the held-input mean error and remains statistically consistent.

Powered ascent in a rotating Earth frame¶

The four-second sounding-rocket scenario includes 12 g measured specific force, changing attitude, gravity, and Earth rotation. “Specified” means the known Earth rate is supplied through omegaCoriolis.

The expected mean-NEES interval is again [8.849, 9.152]. Correctly supplying Earth rate improves every backend. Galilean is the only backend inside the interval when Earth rate is specified.

Long-horizon uncertainty with fixed nonzero bias¶

This ten-second experiment centers every backend on its own noise-free discrete mean, isolating accumulated sensor uncertainty while retaining a fixed nonzero accelerometer and gyroscope bias. The expected mean-NEES interval is [8.815, 9.187].

All four Logmap results are statistically consistent and within 0.007 of the theoretical mean.

Full 15D uncertainty with bias random walk¶

The Combined experiment samples accelerometer and gyroscope bias random walks and retains the full state-bias covariance. Its expected mean-NEES interval is [14.761, 15.241].

Every backend remains inside the confidence interval when the state block uses the Logmap residual.

Right-applied first-order bias correction¶

The companion paper defines the Galilean first-order correction

on the right because its direct-product state uses a standard left-invariant error. The deterministic experiment compares this correction, and each other backend’s corresponding first-order approximation, against complete reintegration. It uses a nonzero bias linearization point, one second of smooth three-axis motion, 2,000 paired random directions per magnitude, and $\lVert\delta b_a\rVert=30\lVert\delta b_\omega\rVert$. The table reports median error at the largest joint update, 0.35.

The corrected Lie-group implementation materially improves the position and velocity approximation over the additive Manifold/Tangent correction. Its velocity error is about seven times smaller at the largest update. Galilean is best in position and velocity for this held-input trajectory. Manifold, Lie-group, and Galilean have nearly identical attitude error, while Tangent’s tangent-space correction differs.

All backends are exact at zero bias update, and their local first-order approximation errors scale quadratically with update magnitude.

Recommendations¶

• Use the default Logmap IMU factor-error mode.

• Use Galilean preintegration for high dynamics, simultaneous rotation and acceleration, lower IMU rates, longer held-input intervals, or rotating navigation frames.

• Use Tangent for established general-purpose and PIM-merging workflows.

• Use Lie group when the formulation requires $SE_2(3)$ increments and its complete right-applied group bias update.

• Supply omegaCoriolis whenever navigation-frame rotation is known.

Legacy and ComponentWise remain compatibility choices, but they are not active modes in the current notebooks.