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On the Optimized Use of Non-Orthonormality Constraints for the Quasi-Static INS Alignment of Autonomous Underwater and Surface Vehicles
Carlos Renato C. Durao, Felipe O. Silva, Itzik Klein, Vinıcius M. G. B. Cavalcanti, Adriano Frutuoso, Ettore A. de Barros, Jay A. Farrell
TL;DR
Quasi-static INS alignment for AUVs and ASVs can suffer from slow convergence and limited bias estimability. The paper introduces OPT-TRIAD-CBE and a NON-constrained EKF observation model, validated in simulations and real-world experiments. The approaches accelerate estimation from minutes to seconds while maintaining accuracy and precision comparable to traditional methods.
Problem
Conventional quasi-static INS alignment methods can converge slowly and weakly estimate some inertial sensor-bias components.
Method
The paper combines Weighted Least Squares optimization of TRIAD-CBE with an EKF Fine Alignment observation model based on TRIAD-derived NON error equations.
Results
The proposed methods substantially accelerate misalignment and bias-estimate convergence from minutes to seconds while retaining accuracy and precision comparable to traditional techniques.
Takeaways & Limitations
OPT-TRIAD-CBE and the NON-based FA provide practical quasi-static alignment improvements for AUVs and ASVs, particularly in convergence rate and estimable-bias estimation.
Takeaways & Limitations
The proposed NON-based FA can perform worse than OPT-TRIAD-CBE in bias-estimation accuracy and convergence, potentially because of noisy terms and process–measurement-noise correlation.
Abstract
from arXiv · showhide
Inertial navigation systems are specialized navigation apparatuses that equip almost all autonomous underwater and surface vehicles. They require precise initial alignment, i.e., determination of their initial attitude, which is typically achieved: (a) in quasi-static conditions (whenever possible); and (b) in two stages: Coarse Alignment (CA), using methods like TRI-axis Attitude Determination (TRIAD), and Fine Alignment (FA), via Zero Velocity Update (ZVU)-based Extended Kalman Filtering (EKF). However, conventional methods suffer from slow convergence and limited bias estimability. In response, this paper introduces: (a) an optimized version of a recently proposed CA method, namely, TRIAD with Coarse Bias Estimation (TRIAD-CBE); and (b) a novel FA EKF observation model that incorporates Non-Orthonormality (NON) error constraints derived from TRIAD, directly linking these errors to the inertial sensor biases. As validated through extensive Monte Carlo (MC) simulations, as well as real-world experiments using two Inertial Measurement Units (IMUs) of different grades, our approaches substantially accelerate the convergence of misalignment and bias estimates (from minutes to seconds), while maintaining accuracy/precision comparable to traditional techniques.
I. INTRODUCTION
Quasi-static INS alignment for AUVs and ASVs uses coarse and fine stages, but conventional methods can converge slowly and estimate some sensor biases weakly. The paper proposes optimized TRIAD-CBE and a NON-based EKF observation model to improve convergence and bias estimation.
- Motivation: AUV/ASV INS alignment is especially relevant because underwater GNSS attenuation makes navigation and localization more challenging for AUVs.INS initialization, particularly attitude alignment, should be accurate, fast, and autonomous under quasi-static conditions.
- Conventional alignment: Quasi-static alignment generally consists of Coarse Alignment using gravity and Earth-rate vectors, followed by Fine Alignment to improve attitude and estimate inertial sensor biases.TRIAD analytically computes an initial attitude from corresponding vector bases, while FA commonly uses an EKF with ZVU measurements.
- Limitations: Traditional ZVU-based FA can weakly estimate some sensor-bias components or require long convergence times.Prior work attempted to accelerate FA with CERGUs, backtracking, analytical quaternion relations, and mixed aiding strategies.
- Contributions: The paper introduces OPT-TRIAD-CBE, which uses Weighted Least Squares for bias estimation, and a NON-based EKF observation model for FA.The proposed FA model reuses NON error equations originally derived for TRIAD-CBE.
- Reported outcomes: The proposed FA model largely improves convergence over traditional ZVU-based EKF, while OPT-TRIAD-CBE improves accuracy, precision, and speed for estimable biases.The reported advantages are especially pronounced when gyroscope noise is high and time-correlated.
II. The Traditional Alignment Method
Traditional quasi-static alignment obtains an initial TRIAD attitude from gravity and Earth-rate observations, then uses a ZVU-based EKF to correct misalignment and estimate biases. Bias-corrupted measurements drive navigation errors, while ZVU supplies the EKF measurement during stationary operation.
- Coarse Alignment: TRIAD determines the body-to-NED rotation matrix from gravity and Earth-rate vectors observed in both frames.The body-frame observations are approximated using quasi-static accelerometer and gyroscope measurements.
- Coarse Alignment: Traditional TRIAD coarse alignment ignores inertial sensor biases, making its rotation-matrix estimate inaccurate and motivating subsequent Fine Alignment.Fine Alignment also estimates sensor biases to improve the attitude estimate.
- Error propagation: Bias-corrupted inertial measurements cause navigation-equation error components to grow over time.An EKF augments sensor biases in the state vector and estimates them under quasi-static conditions.
- EKF model: The EKF error model includes velocity and misalignment errors, accelerometer and gyroscope biases, sensor noises, and Earth-curvature parameters.Accelerometer and gyroscope noises are modeled as white and Gaussian, with process-noise covariance specified separately.
- Fine Alignment: The traditional FA measurement is the INS-computed Earth-referenced velocity error, modeled as zero under a Zero Velocity Update condition.The measurement equation corrects state estimates during the EKF update step.
III. The TRIAD-CBE Method
TRIAD-CBE preserves TRIAD-generated non-orthonormality errors and analytically relates them to estimable combinations of inertial sensor biases. This enables coarse bias estimation before Fine Alignment, although only selected bias combinations are recoverable.
- NON errors: TRIAD produces an estimated rotation matrix whose errors comprise misalignment, normality, and orthogonality components.The normality and orthogonality errors are represented by η and o vectors.
- Bias relationship: TRIAD-CBE uses TRIAD-derived NON errors because their equations relate directly to selected inertial sensor-bias components.The relationship supports coarse bias estimation during the CA procedure.
- Bias estimation: TRIAD-CBE estimates down accelerometer and north/down gyroscope biases when position and gravity are accurately known.Latitude and gravity biases are included in the analytical relationships used for estimation.
- Estimability: Three bias combinations can be estimated during CA, while north/east accelerometer and east gyroscope biases cannot be estimated likewise.The method may accelerate overall CA+FA alignment but does not necessarily improve its accuracy.
IV. The Optimized TRIAD-CBE Method
OPT-TRIAD-CBE extends TRIAD-CBE by modeling noise in TRIAD-generated NON errors and solving the resulting bias-estimation system with a weighted least-squares procedure. The method is characterized as an optimal estimator for the estimable biases and misalignment errors.
- Noise-aware formulation: OPT-TRIAD-CBE adds the noise components corrupting TRIAD-generated NON errors, which were omitted in the original TRIAD-CBE formulation.This modification makes the bias-estimation model account for measurement-noise effects.
- Identifiability: The NON-error system is ill-conditioned because ηE = ηN + ηD creates linearly dependent equations.Consequently, not all bias parameters can be determined directly.
- Linear model: The NON-error equations form a linear system relating observed errors to estimable bias combinations and system noises.The observation vector contains ηN, ηE, ηD, and oE.
- Estimator: OPT-TRIAD-CBE obtains bias estimates with a whitened Weighted Least Squares solution using SVD and a Moore–Penrose pseudo-inverse.The weighting accounts for singularities in the weighting matrix and incorporates system-noise variances.
- Alignment impact: After compensating estimated biases in the raw sensor readings, the method is expected to provide optimal misalignment estimates as well.The optimized estimates target the down accelerometer and north/down gyroscope biases when position and gravity are known accurately.
V. The Proposed Fine Alignment Method
The proposed ZVU+NON fine-alignment model augments the traditional EKF measurement vector with TRIAD-derived non-orthonormality error estimates. Its covariance is represented for the augmented noise vector, and the model is named ZVU+NON.
- ZVU+NON augments the traditional ZVU-based EKF fine-alignment measurement vector with TRIAD-derived NON error estimates.TRIAD is assumed to run in parallel with the fine-alignment process.
- The augmented EKF model uses a covariance matrix specified for the augmented noise vector.
- The augmented EKF-based fine-alignment model is referred to as ZVU+NON.
VI. Simulated Experiments
Monte Carlo simulations compare TRIAD-CBE, OPT-TRIAD-CBE, and EKF-based fine-alignment methods on misalignment, bias estimation, precision, and convergence. Misalignment accuracy is equivalent across methods, while TRIAD-CBE variants improve estimable bias performance and ZVU+NON substantially accelerates convergence.
- Monte Carlo setup: 10,000 statistically independent realizations were used to provide a more reliable assessment than the single-run simulation.The realizations repeated the previous stationary-sensor simulation and retained final misalignment and estimable-bias values.
- Misalignment accuracy: All investigated methods had equivalent misalignment-component estimation performance, and fine alignment did not improve on coarse-alignment accuracy.Analytical standard-deviation expressions agreed with the Monte Carlo distributions.
- Bias accuracy and precision: TRIAD-CBE and OPT-TRIAD-CBE performed best for down accelerometer bias and north gyro bias, while methods were similar for down gyro bias.The comparison concerns both accuracy and precision of the estimated bias error components.
- Bias accuracy and precision: σbaD = 0.0001 mg, σbgN = 0.003 deg/h, and σbgD = 0.007 deg/h were obtained for TRIAD-CBE bias uncertainties.These values agreed with the reported metrics, confirming TRIAD-CBE precision for the estimable bias components.
- Convergence: ZVU+NON reduced estimate convergence from minutes to a few seconds, including relative to the ZVU+CERGU benchmark.Convergence time was defined as stabilization within ±5% of the steady-state value.
- Convergence: TRIAD-CBE and OPT-TRIAD-CBE stabilized all states of interest within average times of 29 and 41 seconds, respectively.
VII. Real-World Experiments
Real-world experiments on a turntable and a stationary ASV reproduced the simulated patterns: comparable attitude accuracy across methods, but substantially faster bias and misalignment convergence for the proposed approaches, especially OPT-TRIAD-CBE.
- Turntable experiment: Turntable experiments showed comparable misalignment accuracy across CA and FA methods, while the proposed methods significantly accelerated misalignment and bias convergence.OPT-TRIAD-CBE was especially effective compared with ZVU- and ZVU-CERGU-based fine alignments.
- ASV experiment: In the ASV experiment, roll and pitch estimates differed by less than 0.01 degrees, while heading differences remained below 0.1 degrees.The estimates converged to closely agreeing attitude values across methods.
- ASV experiment: Within 300 seconds, only TRIAD-CBE and OPT-TRIAD-CBE produced steady-state estimates for the north and down gyroscope biases.Other methods showed only a tendency toward the values estimated by TRIAD-CBE and OPT-TRIAD-CBE.
- Experimental conditions: The ASV gyroscopes had approximately 600 deg/hour noise and residual biases of 0.02 deg/h, complicating fair comparison of bias-estimation performance.The experiment was rerun after adding 1 deg/h biases to the gyroscope signals.
- Modified ASV and Monte Carlo tests: For alignment intervals below 50 seconds, OPT-TRIAD-CBE produced more precise gyroscope-bias estimates than TRIAD-CBE.The difference was reflected in the smaller standard deviations of the estimates.
- Modified ASV and Monte Carlo tests: Across the experiments, OPT-TRIAD-CBE showed better precision under 30-second alignment with increased white and time-correlated gyroscope noise.This modified Monte Carlo setup was designed to reproduce the ASV sensor conditions.
VIII. Conclusion
The paper proposes OPT-TRIAD-CBE and ZVU-NON for quasi-static AUV/ASV alignment, using TRIAD-derived non-orthonormality errors to improve bias estimation and EKF observations. The methods preserve comparable misalignment accuracy while reducing convergence from minutes to seconds, although ZVU-NON remains less precise than the CA alternatives and in-motion use remains future work.
- Contributions: The paper proposes optimized TRIAD-CBE coarse alignment and a NON-error-based EKF observation model for quasi-static AUV/ASV fine alignment.The NON errors originate from ordinary TRIAD-derived rotation matrices and correlate directly with some inertial sensor biases.
- Accuracy: The proposed methods matched baseline methods in misalignment accuracy and precision across Monte Carlo and real-world experiments.The baselines were TRIAD-CBE, ZVU, and ZVU+CERGU.
- Convergence: From minutes to just a few seconds, the proposed approaches accelerated estimation of down misalignment and estimable inertial sensor biases.OPT-TRIAD-CBE and TRIAD-CBE also outperformed ZVU-NON in bias accuracy/precision and convergence rate in the Monte Carlo analysis.
- Limitations and future work: ZVU-NON was less precise because it depends on noisy high-order derivatives of estimable velocity errors.The authors identify possible process–measurement noise correlation as a future research question.