Gauge Freedom Optimization for Truncation Error Reduction in Inertial Navigation

Numerical integration plays a central role in inertial navigation systems, where sensor measurements are propagated through time to obtain orientation, velocity, and position states. The accuracy of this propagation depends on the numerical integrator type, order and step-size. Prior work showed that for second-order systems with known forcing functions, the gauge freedom in the variation of parameters technique can be exploited to reduce truncation error without modifying the integrator. However, this approach requires analytical knowledge of the forcing function, limiting its applicability in real-world systems. To address this limitation we propose the u-space methodology, a novel state mapping that generalizes the gauge freedom to systems with unknown forcing functions. The optimal gauge is derived in closed form for second-order systems and in both closed and empirical form for first-order systems. The proposed approach was evaluated through Monte Carlo simulations across four forcing functions, five sensor grades, and four Adams-Bashforth orders, as well as on a real-world inertial navigation dataset. Results show consistent error reduction across all tested conditions, with the largest gains observed in the full inertial mechanization pipeline, making the approach applicable to high-grade inertial systems, where truncation error constitutes a larger share of the error budget, and to aided low-cost systems with high-rate updates, where propagation spans only short inter-update intervals.

A Unified Filter for Fusion of Multiple Inertial Measurement Units

Navigation plays a vital role in the ability of autonomous surface and underwater platforms to complete their tasks. Most navigation systems apply a fusion between inertial sensors and other external sensors, such as global navigation satellite systems, when available, or a Doppler velocity log. In recent years, there has been increased interest in using multiple inertial measurement units to improve navigation accuracy and robustness. State of the art examples include the virtual inertial measurement unit (VIMU) and the federated extended Kalman filter (FEKF). However, each of those approaches has its drawbacks. The VIMU does not improve the sensor biases, which constitute a significant source of error, especially in low-cost inertial sensors. While the FEKF does improve accuracy, it models uncertainty propagation empirically. If not modeled correctly, this can cause the global solution to diverge. To cope with those shortcomings, we propose and derive a new filter structure for multiple inertial sensors data fusion: the unified extended Kalman filter (UEKF). In addition, to cope with the multiple equal bias variance estimation problem we offer the bias variance redistribution algorithm. Our filter design enables bias estimation for each of the inertial sensors in the system, improving its accuracy and allowing the use of a varying number of inertial sensors. We show that our UEKF performs better than other state of the art, multiple inertial sensor filters using real data recorded during sea experiments.