Yuanxin Wu

Yuanxin Wu, Hongliang Zhang, Meiping Wu et al.

Alignment of the strapdown inertial navigation system (INS) has strong nonlinearity, even worse when maneuvers, e.g., tumbling techniques, are employed to improve the alignment. There is no general rule to attack the observability of a nonlinear system, so most previous works addressed the observability of the corresponding linearized system by implicitly assuming that the original nonlinear system and the linearized one have identical observability characteristics. Strapdown INS alignment is a nonlinear system that has its own characteristics. Using the inherent properties of strapdown INS, e.g., the attitude evolution on the SO(3) manifold, we start from the basic definition and develop a global and constructive approach to investigate the observability of strapdown INS static and tumbling alignment, highlighting the effects of the attitude maneuver on observability. We prove that strapdown INS alignment, considering the unknown constant sensor biases, will be completely observable if the strapdown INS is rotated successively about two different axes and will be nearly observable for finite known unobservable states (no more than two) if it is rotated about a single axis. Observability from a global perspective provides us with insights into and a clearer picture of the problem, shedding light on previous theoretical results on strapdown INS alignment that were not comprehensive or consistent.. The reporting of inconsistencies calls for a review of all linearization-based observability studies in the vast literature. Extensive simulations with constructed ideal observers and an extended Kalman filter are carried out, and the numerical results accord with the analysis. The conclusions can also assist in designing the optimal tumbling strategy and the appropriate state observer in practice to maximize the alignment performance.

Xinrui Li, Qi Cai, Yuanxin Wu

Structure from Motion (SfM) is a critical task in computer vision, aiming to recover the 3D scene structure and camera motion from a sequence of 2D images. The recent pose-only imaging geometry decouples 3D coordinates from camera poses and demonstrates significantly better SfM performance through pose adjustment. Continuing the pose-only perspective, this paper explores the critical relationship between the scene structures, rotation and translation. Notably, the translation can be expressed in terms of rotation, allowing us to condense the imaging geometry representation onto the rotation manifold. A rotation-only optimization framework based on reprojection error is proposed for both two-view and multi-view scenarios. The experiment results demonstrate superior accuracy and robustness performance over the current state-of-the-art rotation estimation methods, even comparable to multiple bundle adjustment iteration results. Hopefully, this work contributes to even more accurate, efficient and reliable 3D visual computing.

Yuqi Han, Qi Cai, Yuanxin Wu

Offline camera calibration techniques typically employ parametric or generic camera models. Selecting parametric models relies heavily on user experience, and an inappropriate camera model can significantly affect calibration accuracy. Meanwhile, generic calibration methods involve complex procedures and cannot provide traditional intrinsic parameters. This paper reveals a pose ambiguity in the pose solutions of generic calibration methods that irreversibly impacts subsequent pose estimation. A linear solver and a nonlinear optimization are proposed to address this ambiguity issue. Then a global optimization hybrid calibration method is introduced to integrate generic and parametric models together, which improves extrinsic parameter accuracy of generic calibration and mitigates overfitting and numerical instability in parametric calibration. Simulation and real-world experimental results demonstrate that the generic-parametric hybrid calibration method consistently excels across various lens types and noise contamination, hopefully serving as a reliable and accurate solution for camera calibration in complex scenarios.

Qi Cai, Xinrui Li, Yuanxin Wu

How to efficiently and accurately handle image matching outliers is a critical issue in two-view relative estimation. The prevailing RANSAC method necessitates that the minimal point pairs be inliers. This paper introduces a linear relative pose estimation algorithm for n $( n \geq 6$) point pairs, which is founded on the recent pose-only imaging geometry to filter out outliers by proper reweighting. The proposed algorithm is able to handle planar degenerate scenes, and enhance robustness and accuracy in the presence of a substantial ratio of outliers. Specifically, we embed the linear global translation (LiGT) constraint into the strategies of iteratively reweighted least-squares (IRLS) and RANSAC so as to realize robust outlier removal. Simulations and real tests of the Strecha dataset show that the proposed algorithm achieves relative rotation accuracy improvement of 2 $\sim$ 10 times in face of as large as 80% outliers.

Maoran Zhu, Yuanxin Wu

In this paper, a new framework for continuous-time maximum a posteriori estimation based on the Chebyshev polynomial optimization (ChevOpt) is proposed, which transforms the nonlinear continuous-time state estimation into a problem of constant parameter optimization. Specifically, the time-varying system state is represented by a Chebyshev polynomial and the unknown Chebyshev coefficients are optimized by minimizing the weighted sum of the prior, dynamics and measurements. The proposed ChevOpt is an optimal continuous-time estimation in the least squares sense and needs a batch processing. A recursive sliding-window version is proposed as well to meet the requirement of real-time applications. Comparing with the well-known Gaussian filters, the ChevOpt better resolves the nonlinearities in both dynamics and measurements. Numerical results of demonstrative examples show that the proposed ChevOpt achieves remarkably improved accuracy over the extended/unscented Kalman filters and extended batch/fixed-lag smoother, closes to the Cramer-Rao lower bound.

Yuanxin Wu, Maoran Zhu

Time-equispaced inertial measurements are practically used as inputs for motion determination. Polynomial interpolation is a common technique of recovering the gyroscope signal but is subject to a fundamentally numerical stability problem due to the Runge effect on equispaced samples. This paper reviews the theoretical results of Runge phenomenon in related areas and proposes a straightforward borrowing-and-cutting (BAC) strategy to depress it. It employs the neighboring samples for higher-order polynomial interpolation but only uses the middle polynomial segment in the actual time interval. The BAC strategy has been incorporated into attitude computation by functional iteration, leading to accuracy benefit of several orders of magnitude under the classical coning motion. It would potentially bring significant benefits to the inertial navigation computation under sustained dynamic motions.

Qi Cai, Lilian Zhang, Yuanxin Wu et al.

Visual navigation and three-dimensional (3D) scene reconstruction are essential for robotics to interact with the surrounding environment. Large-scale scenes and critical camera motions are great challenges facing the research community to achieve this goal. We raised a pose-only imaging geometry framework and algorithms that can help solve these challenges. The representation is a linear function of camera global translations, which allows for efficient and robust camera motion estimation. As a result, the spatial feature coordinates can be analytically reconstructed and do not require nonlinear optimization. Experiments demonstrate that the computational efficiency of recovering the scene and associated camera poses is significantly improved by 2-4 orders of magnitude. This solution might be promising to unlock real-time 3D visual computing in many forefront applications.

Wei Ouyang, Yuanxin Wu

This work deals with error models for trident quaternion framework proposed in the companion paper (Part I) and further uses them to investigate the odometer-aided static/in-motion inertial navigation attitude alignment for land vehicles. By linearizing the trident quaternion kinematic equation, the left and right trident quaternion error models are obtained, which are found to be equivalent to those derived from profound group affine. The two error models are used to design their corresponding extended Kalman filters (EKF), namely, the left-quaternion EKF (LQEKF) and the right-quaternion EKF (RQEKF). Simulations and field tests are conducted to evaluate their actual performances. Owing to the high estimation consistency, the L/RQEKF converge much faster in the static alignment than the traditional error model-based EKF, even under arbitrary large heading initialization. For the in-motion alignment, the L/RQEKF possess much larger convergence region than the traditional EKF does, although they still require the aid of attitude initialization so as to avoid large initial attitude errors.

Wei Ouyang, Yuanxin Wu

Strapdown inertial navigation research involves the parameterization and computation of the attitude, velocity and position of a rigid body in a chosen reference frame. The community has long devoted to finding the most concise and efficient representation for the strapdown inertial navigation system (INS). The current work is motivated by simplifying the existing dual quaternion representation of the kinematic model. This paper proposes a compact and elegant representation of the body's attitude, velocity and position, with the aid of a devised trident quaternion tool in which the position is accounted for by adding a second imaginary part to the dual quaternion. Eventually, the kinematics of strapdown INS are cohesively unified in one concise differential equation, which bears the same form as the classical attitude quaternion equation. In addition, the computation of this trident quaternion-based kinematic equation is implemented with the recently proposed functional iterative integration approach. Numerical results verify the analysis and show that incorporating the new representation into the functional iterative integration scheme achieves high inertial navigation computation accuracy as well.

Maoran Zhu, Yuanxin Wu, Shitu Luo

Foot-mounted inertial sensors become popular in many indoor or GPS-denied applications, including but not limited to medical monitoring, gait analysis, soldier and first responder positioning. However, the foot-mounted inertial navigation relies largely on the aid of Zero Velocity Update (ZUPT) and has encountered inherent problems such as heading drift. This paper implements a pedestrian navigation system based on dual foot-mounted low-cost inertial measurement units (IMU) and inter-foot ultrasonic ranging. The observability analysis of the system is performed to investigate the roles of the ZUPT measurement and the foot-to-foot ranging measurement in improving the state estimability. A Kalman-based estimation algorithm is mechanized in the Earth frame, rather than in the common local-level frame, which is found to be effective in depressing the linearization error in Kalman filtering. An ellipsoid constraint in the Earth frame is also proposed to further restrict the height drift. Simulation and real field experiments show that the proposed method has better robustness and positioning accuracy (about 0.1-0.2% travelled distance) than the traditional pedestrian navigation schemes do.

Shengjie Li, Qi Cai, Yuanxin Wu

Identifying feature correspondence between two images is a fundamental procedure in three-dimensional computer vision. Usually the feature search space is confined by the epipolar line. Using the cheirality constraint, this paper finds that the feature search space can be restrained to one of two or three segments of the epipolar line that are defined by the epipole and a so-called virtual infinity point.

Wei Ouyang, Yuanxin Wu, Hongyue Chen

Land vehicle navigation based on inertial navigation system (INS) and odometers is a classical autonomous navigation application and has been extensively studied over the past several decades. In this work, we seriously analyze the error characteristics of the odometer (OD) pulses and investigate three types of odometer measurement models in the INS/OD integrated system. Specifically, in the pulse velocity model, a preliminary Kalman filter is designed to obtain accurate vehicle velocity from the accumulated pulses; the pulse increment model is accordingly obtained by integrating the pulse velocity; a new pulse accumulation model is proposed by augmenting the travelled distance into the system state. The three types of measurements, along with the nonhonolomic constraint (NHC), are implemented in the standard extended Kalman filter. In view of the motion-related pulse error characteristics, the multiple model adaptive estimation (MMAE) approach is exploited to further enhance the performance. Simulations and long-distance experiments are conducted to verify the feasibility and effectiveness of the proposed methods. It is shown that the standard pulse velocity measurement achieves the superior performance, whereas the accumulated pulse measurement is most favorable with the MMAE enhancement.

Yuanxin Wu, Yury A. Litmanovich

This paper compares two basic approaches to solving ordinary differential equations, which form the basis for attitude computation in strapdown inertial navigation systems, namely, the Taylor series expansion approach that was used in its low-order form for deriving all mainstream algorithms and the functional iterative integration approach developed recently. They are respectively applied to solve the kinematic equations of major attitude parameters, including the quaternion, the Rodrigues vector and the rotation vector. Specifically, the mainstream algorithms, which have relied on the simplified rotation vector without exception, are considerably extended by the Taylor series expansion approach using the exact rotation vector and recursive calculation of high-order derivatives. The functional iterative integration approach is respectively implemented on both the normal polynomial and the Chebyshev polynomial. Numerical results under the classical coning motion are reported to assess all derived attitude algorithms. It is revealed that in the relative frequency range when the coning to sampling frequency ratio is below 0.05-0.1 (depending on the chosen polynomial truncation order), all algorithms have the same order of accuracy if the same number of samples are used to fit the angular velocity over the iteration interval; in the range of higher relative frequency, the group of Quat/Rod/RotFIter algorithms (by the functional iterative integration approach combined with the Chebyshev polynomial) perform the best in both accuracy and robustness, thanks to the excellent numerical stability and powerful functional representation capability of the Chebyshev polynomial.

Yuanxin Wu

Inertial navigation computation is to acquire the attitude, velocity and position information of a moving body by integrating inertial measurements from gyroscopes and accelerometers. Over half a century has witnessed great efforts in coping with the motion non-commutativity errors to accurately compute the navigation information as far as possible, so as not to compromise the quality measurements of inertial sensors. Highly dynamic applications and the forthcoming cold-atom precision inertial navigation systems demand for even more accurate inertial navigation computation. The paper gives birth to an inertial navigation algorithm to fulfill that demand, named the iNavFIter, which is based on a brand-new framework of functional iterative integration and Chebyshev polynomials. Remarkably, the proposed iNavFIter reduces the non-commutativity errors to almost machine precision, namely, the coning/sculling/scrolling errors that have perplexed the navigation community for long. Numerical results are provided to demonstrate its accuracy superiority over the state-of-the-art inertial navigation algorithms at affordable computation cost.

Qi Cai, Tsung-Ching Lin, Yuanxin Wu et al.

A general and fast method is conceived for computing the cyclic convolution of n points, where n is a prime number. This method fully exploits the internal structure of the cyclic matrix, and hence leads to significant reduction of the multiplication complexity in terms of CPU time by 50%, as compared with Winograd's algorithm. In this paper, we only consider the real and complex fields due to their most important applications, but in general, the idea behind this method can be extended to any finite field of interest. Clearly, it is well-known that the discrete Fourier transform (DFT) can be expressed in terms of cyclic convolution, so it can be utilized to compute the DFT when the block length is a prime.

Yuanxin Wu, Chao He, Gang Liu

Inertial navigation and attitude initialization in polar areas become a hot topic in recent years in the navigation community, as the widely-used navigation mechanization of the local level frame encounters the inherent singularity when the latitude approaches 90 degrees. Great endeavors have been devoted to devising novel navigation mechanizations such as the grid or transversal frames. This paper highlights the fact that the common Earth-frame mechanization is sufficiently good to well handle the singularity problem in polar areas. Simulation results are reported to demonstrate the singularity problem and the effectiveness of the Earth-frame mechanization.

Yuanxin Wu, Gongmin Yan

RodFIter is a promising method of attitude reconstruction from inertial measurements based on the functional iterative integration of Rodrigues vector. The Rodrigues vector is used to encode the attitude in place of the popular rotation vector because it has a polynomial-like rate equation and could be cast into theoretically sound and exact integration. This paper further applies the approach of RodFIter to the unity-norm quaternion for attitude reconstruction, named QuatFIter, and shows that it is identical to the previous Picard-type quaternion method. The Chebyshev polynomial approximation and truncation techniques from the RodFIter are exploited to speed up its implementation. Numerical results demonstrate that the QuatFIter is comparable in accuracy to the RodFIter, although its convergence rate is relatively slower with respect to the number of iterations. Notably, the QuatFIter has about two times better computational efficiency, thanks to the linear quaternion kinematic equation.

Danping Zou, Yuanxin Wu, Ling Pei et al.

We propose a novel visual-inertial odometry approach that adopts structural regularity in man-made environments. Instead of using Manhattan world assumption, we use Atlanta world model to describe such regularity. An Atlanta world is a world that contains multiple local Manhattan worlds with different heading directions. Each local Manhattan world is detected on-the-fly, and their headings are gradually refined by the state estimator when new observations are coming. With fully exploration of structural lines that aligned with each local Manhattan worlds, our visual-inertial odometry method become more accurate and robust, as well as much more flexible to different kinds of complex man-made environments. Through extensive benchmark tests and real-world tests, the results show that the proposed approach outperforms existing visual-inertial systems in large-scale man-made environments

Qi Cai, Yuanxin Wu, Lilian Zhang et al.

Two-view relative pose estimation and structure reconstruction is a classical problem in computer vision. The typical methods usually employ the singular value decomposition of the essential matrix to get multiple solutions of the relative pose, from which the right solution is picked out by reconstructing the three-dimension (3D) feature points and imposing the constraint of positive depth. This paper revisits the two-view geometry problem and discovers that the two-view imaging geometry is equivalently governed by a Pair of new Pose-Only (PPO) constraints: the same-side constraint and the intersection constraint. From the perspective of solving equation, the complete pose solutions of the essential matrix are explicitly derived and we rigorously prove that the orientation part of the pose can still be recovered in the case of pure rotation. The PPO constraints are simplified and formulated in the form of inequalities to directly identify the right pose solution with no need of 3D reconstruction and the 3D reconstruction can be analytically achieved from the identified right pose. Furthermore, the intersection inequality also enables a robust criterion for pure rotation identification. Experiment results validate the correctness of analyses and the robustness of the derived pose solution/pure rotation identification and analytical 3D reconstruction.

Yuanxin Wu, Qi Cai, Tnieu-Kien Truong

Attitude computation is of vital importance for a variety of applications. Based on the functional iteration of the Rodrigues vector integration equation, the RodFIter method can be advantageously applied to analytically reconstruct the attitude from discrete gyroscope measurements over the time interval of interest. It is promising to produce ultra-accurate attitude reconstruction. However, the RodFIter method imposes high computational load and does not lend itself to onboard implementation. In this paper, a fast approach to significantly reduce RodFIter's computation complexity is presented while maintaining almost the same accuracy of attitude reconstruction. It reformulates the Rodrigues vector iterative integration in terms of the Chebyshev polynomial iteration. Due to the excellent property of Chebyshev polynomials, the fast RodFIter is achieved by means of appropriate truncation of Chebyshev polynomials, with provably guaranteed convergence. Moreover, simulation results validate the speed and accuracy of the proposed method.

Yuanxin Wu

Rigid motion computation or estimation is a cornerstone in numerous fields. Attitude computation can be achieved by integrating the angular velocity measured by gyroscopes, the accuracy of which is crucially important for the dead-reckoning inertial navigation. The state-of-the-art attitude algorithms have unexceptionally relied on the simplified differential equation of the rotation vector to obtain the attitude. This paper proposes a Functional Iteration technique with the Rodrigues vector (named the RodFIter method) to analytically reconstruct the attitude from gyroscope measurements. The RodFIter method is provably exact in reconstructing the incremental attitude as long as the angular velocity is exact. Notably, the Rodrigues vector is analytically obtained and can be used to update the attitude over the considered time interval. The proposed method gives birth to an ultimate attitude algorithm scheme that can be naturally extended to the general rigid motion computation. It is extensively evaluated under the attitude coning motion and compares favorably in accuracy with the mainstream attitude algorithms. This work is believed having eliminated the long-standing theoretical barrier in exact motion integration from inertial measurements.

Yuanxin Wu, Ling Pei

Magnetometers, gyroscopes and accelerometers are commonly used sensors in a variety of applications. The paper proposes a novel gyroscope calibration method in the homogeneous magnetic field by the help of magnetometer. It is shown that, with sufficient rotation excitation, the homogeneous magnetic field vector can be exploited to serve as a good reference for calibrating low-cost gyroscopes. The calibration parameters include the gyroscope scale factor, non-orthogonal coefficient and bias for three axes, as well as its misalignment to the magnetometer frame. Simulation and field test results demonstrate the method's effectiveness.

Yuanxin Wu, Danping Zou, Peilin Liu et al.

Magnetometer and inertial sensors are widely used for orientation estimation. Magnetometer usage is often troublesome, as it is prone to be interfered by onboard or ambient magnetic disturbance. The onboard soft-iron material distorts not only the magnetic field, but the magnetometer sensor frame coordinate and the cross-sensor misalignment relative to inertial sensors. It is desirable to conveniently put magnetic and inertial sensors information in a common frame. Existing methods either split the problem into successive intrinsic and cross-sensor calibrations, or rely on stationary accelerometer measurements which is infeasible in dynamic conditions. This paper formulates the magnetometer calibration and alignment to inertial sensors as a state estimation problem, and collectively solves the magnetometer intrinsic and cross-sensor calibrations, as well as the gyroscope bias estimation. Sufficient conditions are derived for the problem to be globally observable, even when no accelerometer information is used at all. An extended Kalman filter is designed to implement the state estimation and comprehensive test data results show the superior performance of the proposed approach. It is immune to acceleration disturbance and applicable potentially in any dynamic conditions.

Xianfei Pan, Yuanxin Wu

Precise autonomous navigation remains a substantial challenge to all underwater platforms. Inertial Measurement Units (IMU) and Doppler Velocity Logs (DVL) have complementary characteristics and are promising sensors that could enable fully autonomous underwater navigation in unexplored areas without relying on additional external Global Positioning System (GPS) or acoustic beacons. This paper addresses the combined IMU/DVL navigation system from the viewpoint of observability. We show by analysis that under moderate conditions the combined system is observable. Specifically, the DVL parameters, including the scale factor and misalignment angles, can be calibrated in-situ without using external GPS or acoustic beacon sensors. Simulation results using a practical estimator validate the analytic conclusions.

Yuanxin Wu, Wei Shi

Magnetometer has received wide applications in attitude determination and scientific measurements. Calibration is an important step for any practical magnetometer use. The most popular three-axis magnetometer calibration methods are attitude-independent and have been founded on an approximate maximum likelihood (ML) estimation with a quartic subjective function, derived from the fact that the magnitude of the calibrated measurements should be constant in a homogeneous magnetic field. This paper highlights the shortcomings of those popular methods and proposes to use the quadratic optimal ML estimation instead for magnetometer calibration. Simulation and test results show that the optimal ML calibration is superior to the approximate ML methods for magnetometer calibration in both accuracy and stability, especially for those situations without sufficient attitude excitation. The significant benefits deserve the moderately increased computation burden. The main conclusion obtained in the context of magnetometer in this paper is potentially applicable to various kinds of three-axis sensors.

Yuanxin Wu

Inertial measurement unit (IMU) and odometer have been commonly-used sensors for autonomous land navigation in the global positioning system (GPS)-denied scenarios. This paper systematically proposes a versatile strategy for self-contained land vehicle navigation using the IMU and an odometer. Specifically, the paper proposes a self-calibration and refinement method for IMU/odometer integration that is able to overcome significant variation of the misalignment parameters, which are induced by many inevitable and adverse factors such as load changing, refueling and ambient temperature. An odometer-aided IMU in-motion alignment algorithm is also devised that enables the first-responsive functionality even when the vehicle is running freely. The versatile strategy is successfully demonstrated and verified via long-distance real tests.

Yuanxin Wu, Jinling Wang, Dewen Hu

Integration of inertial navigation system (INS) and global navigation satellite system (GNSS) is usually implemented in engineering applications by way of Kalman-like filtering. This form of INS/GNSS integration is prone to attitude initialization failure, especially when the host vehicle is moving freely. This paper proposes an online constrained-optimization method to simultaneously estimate the attitude and other related parameters including GNSS antenna's lever arm and inertial sensor biases. This new technique benefits from self-initialization in which no prior attitude or sensor measurement noise information is required. Numerical results are reported to validate its effectiveness and prospect in high accurate INS/GNSS applications.

Yuanxin Wu

The above-mentioned work [1] in IEEE-TR'08 presented an extended Kalman filter for calibrating the misalignment between a camera and an IMU. As one of the main contributions, the locally weakly observable analysis was carried out using Lie derivatives. The seminal paper [1] is undoubtedly the cornerstone of current observability work in SLAM and a number of real SLAM systems have been developed on the observability result of this paper, such as [2, 3]. However, the main observability result of this paper [1] is founded on an incorrect proof and actually cannot be acquired using the local observability technique therein, a fact that is apparently not noticed by the SLAM community over a number of years.

Yuanxin Wu

This note improves our above-mentioned recent work by effectively depressing the adverse effect of the lever arm on attitude estimation.

Yuanxin Wu, Xianfei Pan

Inertial navigation applications are usually referenced to a rotating frame. Consideration of the navigation reference frame rotation in the inertial navigation algorithm design is an important but so far less seriously treated issue, especially for ultra-high-speed flying aircraft or the future ultra-precision navigation system of several meters per hour. This paper proposes a rigorous approach to tackle the issue of navigation frame rotation in velocity/position computation by use of the newly-devised velocity/position integration formulae in the Part I companion paper. The two integration formulae set a well-founded cornerstone for the velocity/position algorithms design that makes the comprehension of the inertial navigation computation principle more accessible to practitioners, and different approximations to the integrals involved will give birth to various velocity/position update algorithms. Two-sample velocity and position algorithms are derived to exemplify the design process. In the context of level-flight airplane examples, the derived algorithm is analytically and numerically compared to the typical algorithms existing in the literature. The results throw light on the problems in existing algorithms and the potential benefits of the derived algorithm.

Yuanxin Wu, Xianfei Pan

The in-flight alignment is a critical stage for airborne INS/GPS applications. The alignment task is usually carried out by the Kalman filtering technique that necessitates a good initial attitude to obtain satisfying performance. Due to the airborne dynamics, the in-flight alignment is much difficult than alignment on the ground. This paper proposes an optimization-based coarse alignment approach using GPS position/velocity as input, founded on the newly-derived velocity/position integration formulae. Simulation and flight test results show that, with the GPS lever arm well handled, it is potentially able to yield the initial heading up to one degree accuracy in ten seconds. It can serve as a nice coarse in-flight alignment without any prior attitude information for the subsequent fine Kalman alignment. The approach can also be applied to other applications that require aligning the INS on the run.