Lie group error state representation and analysis in earth-centred earth-fixed coordinate system

To address navigation accuracy degradation caused by error accumulation in traditional Strapdown Inertial Navigation Systems (SINS), this paper systematically investigates error state representation and analysis methods for SINS in the earth-centered earth-fixed (ECEF) frame based on Lie group theory. The SINS mechanization in the ECEF frame was established, embedding attitude, speed, and position states into the extended special Euclidean group, and deriving both left-invariant and right-invariant error representations for non-group-affine and group-affine Lie group error models. By introducing auxiliary speed vectors, a SINS mechanization satisfying group-affine dynamic characteristics was constructed. Given the time-varying characteristics of measurement noise in SINS/DVL integrated navigation, the Sage-Husa Adaptive Kalman Filter (AKF) was employed for real-time estimation and adjustment of measurement noise. Five algorithms were compared through shipboard experiments: traditional error model (ESINS), non-group-affine left error model (LSE), non-group-affine right error model (RSE), group-affine left error model (LSEGA), and group-affine right error model (RSEGA). Experimental results showed that under small misalignment angles (e.g., [0.1° 0.1° 0.1°] and [1° 1° 1°]), RSEGA-AKF and RSE-AKF perform comparably to the other algorithms, with only marginal differences in yaw and position accuracy. Their distinct and measurable superiority is exclusively demonstrated under large misalignment angle scenarios. Under [0.1° 0.1° 0.1°] misalignment, yaw RMSE and MAE differ by only 1 × 10 −3 ◦10 −30 , while position RMSE and MAE differences are within 0.7 m and 0.5 m respectively, which are negligible for practical navigation. Under large misalignment angles ([10° 10° 10°] and [30° 30° 30°]), however, the advantages of RSEGA-AKF and RSE-AKF become prominent and clearly measurable: under [30° 30° 30°] misalignment, yaw RMSE and MAE are superior by up to 0.3° and 0.1° respectively, while position RMSE and MAE are reduced by up to 15 m.