Baiqing Hu

Transformed Unscented Kalman Filter

This technical note concerns the deterministic sampling points construction strategy for unscented Kalman filter (UKF) and cubature Kalman filter (CKF). From the numerical-integration viewpoint, a new deterministic sampling points set is derived by orthogonal transformation on the cubature points. By embedding these points into the UKF framework, a modified nonlinear filter named transformed unscented Kalman filter (TUKF) is derived. The TUKF can address the nonlocal sampling problem inherent in CKF while maintaining the virtue of numerical stability for high dimensional problems. Moreover, the methodology proposed in this technical note can be used to construct nonlinear filters with improved accuracy for certain problems. The performance of the proposed algorithm is demonstrated through a nonlinear high dimensional problem.

Multiple Outliers Suppression Derivative-Free Filter Based on Unscented Transformation

D UE to its familiarity and computational feasibility, the Kalman filter (KF) has found numerous applications in automatic control, navigation, and communications since its introduction [1–3]. Many nonlinear Kalman type filters (KTFs) have been introduced to extend the KF to nonlinear dynamic and measurement models by forming a Gaussian approximation to the posterior state distribution. Among these nonlinear KTFs, the unscented Kalman filter (UKF) is most celebrated due to its easy implementation, appropriate performance and computational feasibility [4]. However, it is well documented that when the state or measurement noises are contaminated by outliers, theKTF’s performance can severely degrade, because they rely on weighted least-squares (WLS) criteria which is susceptible to outliers [5]. To handle these outliers, several algorithms based on the concept of robust statistics have been proposed. By minimizing the worst-case estimation error averaged over all samples, theH1 basedKF can be used to treat process noises, measurement noises, and model uncertainties [6]. However, it breaks down in the presence of randomly occurring outliers since the design matrices of the H1 filter cannot accommodate well the outliers induced by the thick tails of a noise distribution [7]. Other approaches are robust to either state or measurement outliers and they can not cope with both types of outliers jointly [8–10]. Therefore, they may yield unreliable results when state andmeasurement outliers occur simultaneously. Converting the classical recursive approach into a batch-mode regression form and solving it iteratively, Gandhi proposed a generalized maximum likelihood (GM) type KF which is robust to both the state and measurement outliers [7]. Through auxiliary unknown variables that are jointly estimated along with the state based on the least-squares criterion regularized with the l1 norm, both the state and measurement outliers can also be handled in [11]. However, these methods of [7,11] are both limited to the linear case. Although Gandhi has extended his method to the nonlinear case making use of the extended Kalman filter (EKF) [12], the crude approximation and the cumbersome derivation and evaluation of Jacobian matrices in the EKF may degrade its performance. References [13–16] have combined the derivative-free filters with the M estimator mainly to handle the measurement outliers. Although the structure proposed in [13–16] can be used to handle the state outliers with many iterations, unfortunately, only one iteration is suggested in [13–16], which is not enough to suppress the state outliers. Moreover, in [13–16] the nonlinear measurement functions are also statistically linearized and haven’t been used directly. To the best knowledge of the authors, robust derivative-free filters without linear or statistically linear approximation that addresses both the state and measurement outliers haven’t been studied before. This Note will focus on the robust state estimation problem with outliers using the UKF framework. Based on the GM perspective on the KF, the quadratic cost function in the KF framework is modified by the robust cost function [9,13], to robustify the KF. In this respect, the robust cost function is virtually used to reformulate the predicted state covariance and the measurement noise covariance. Then the reformulated covariance is propagated through the UKF. To handle the state outliers, the measurement update of the UKF should be iterated, which is accomplished by a modification of the iterated UKF [17]. The rest of this Note is organized as follows. Section II presents a GM perspective on the KF and points out how the robustM estimate methodology can be embedded into the UKF framework without linear or statistically linear approximation. Section III is devoted to derive the robust derivative-free filter called the outlier robust unscented Kalman filter (ORUKF) algorithm. Some discussions of the proposed algorithm and comparisonswith existing algorithms are the subject of Sec. IV. Simulation results and comparisons are presented in Sec. V. Finally, conclusions are drawn in Sec. VI.

Unscented type kalman filter: limitation and combination

This study investigates the shortcomings of existing unscented type Kalman filters (UTKFs) used for state estimation problem of non-linear stochastic dynamic systems. There are three kinds of UTKFs - the traditional unscented Kalman filter, the cubature Kalman filter and the transformed unscented Kalman filter. It was demonstrated in the past that these algorithms could capture the posterior mean and covariance accurately to the second order for any non-linearity when propagated through the true non-linear system. However, they yield different information on the higher order terms. Owing to the dualistic effect of higher order terms on the performance of these algorithms, it is desirable to come up with some solution to preserve the positive effect of this information in a single algorithm. Based on the assumption that the state measurements are Guassian, two methods are proposed in this work as the suitable candidates. The first one is an adaptive method that chooses the UTKF achieving the highest value of the likelihood function. The second method treats these algorithms as sub-filters and uses the partitioning approach to obtain an overall estimate. The numerical simulations show that the proposed methods can have comparable performances as one of the best UTKFs for the problems under consideration.

A Variational Bayesian-Based Unscented Kalman Filter With Both Adaptivity and Robustness

This paper proposes a modified unscented Kalman filter (UKF) with both adaptivity and robustness. In the proposed filter, the adaptivity is achieved by estimating the time-varying measurement noise covariance based on variational Bayesian (VB) approximation. The robustness is achieved by modifying the filter update based on Hubers M-estimation and Gaussian-Newton iterated method. In Gaussian assumptions, the proposed filter has a comparable filtering accuracy with the original UKF and better filtering consistency. When the measurement noise covariance is time-varying and there are outliers in the measurements, the proposed filter can outperform UKF and other adaptive or robust filters (such as VB-based UKF and Huber-based UKF) in terms of both filter accuracy and consistency. The efficacy of the proposed filter is demonstrated through the numerical simulation test and integrated navigation shipborne test.

Comparison of direct navigation mode and indirect navigation mode for integrated SINS/GPS

The integrated strapdown inertial navigation system/global position system (SINS/GPS) is a widely used procedure for position, velocity and attitude determination. There are two options for the state estimation procedure by a filtering system model and the integration filter, namely direct navigation mode that estimates the navigation parameters and indirect navigation mode that estimates the error of navigation parameters with a feedback calculation process. To a certain extent, they both have their own advantages. However, determining which one of the two navigation modes will be more celebrated than the other under different conditions is a beneficial but thus far less seriously treated issue. In this study, two navigation modes are systematically studied in terms of the filtering system model, filtering algorithm and application issues. The simulation test and car-mounted experiments for low-precision and high-precision SINS are carried out to compare the performance of two navigation modes under different conditions. The results prove that direct navigation mode outperforms indirect navigation mode in terms of estimation accuracy and stability for poor conditions, but for good conditions, the advantages of direct navigation mode are compromised due to the large computational burden. Therefore, this study is also expected to facilitate the selection of the appropriate navigation model in practice under different conditions.

Robust square-root cubature Kalman filter based on Huber’s M-estimation methodology

In practical engineering applications, the performance of the standard cubature Kalman filter (CKF) and its square-root version can be severely degraded due to outliers in measurement or contaminated distribution. In order to address the problem, a robust version of CKF is presented using Huber’s M-estimation methodology and square-root filtering framework. By making use of the Huber technique to reformulate the measurement update of CKF in square-root filtering framework, the proposed filter can exhibit robustness and numerical stability against deviation from Gaussian distribution assumption. In simulation tests, four versions of CKF—the standard, the square-root, Huber-based, and the proposed are evaluated in terms of estimation accuracy, numerical stability, and robustness under Gaussian and non-Gaussian distribution. The results are concluded that the square-root version outperforms the others under Gaussian distribution, whereas the proposed filter has improved performance in maintaining the robustness and numerical stability under non-Gaussian distribution. The investigated robust framework can be extended to other Gaussian filtering algorithms and the study is expected to facilitate applications of CKF in practical engineering as well.

Online Estimation of ARW Coefficient of Fiber Optic Gyro

As a standard method for noise analysis of fiber optic gyro (FOG), Allan variance has too large offline computational burden and data storages to be applied to online estimation. To overcome the barriers, the state space model is firstly established for FOG. Then the Sage-husa adaptive Kalman filter (SHAKF) is introduced in this field. Through recursive calculation of measurement noise covariance matrix, SHAKF can avoid the storage of large amounts of history data. However, the precision and stability of this method are still the primary matters that needed to be addressed. Based on this point, a new online method for estimation of the coefficient of angular random walk is proposed. In the method, estimator of measurement noise is constructed by the recursive form of Allan variance at the shortest sampling time. Then the estimator is embedded into the SHAKF framework resulting in a new adaptive filter. The estimations of measurement noise variance and Kalman filter are independent of each other in this method. Therefore, it can address the problem of filtering divergence and precision degrading effectively. Test results of both digital simulation and experimental data of FOG verify the validity and feasibility of the proposed method.

Comment on “Highly Efficient Sigma Point Filter for Spacecraft Attitude and Rate Estimation”

In light of the intuition that a better symmetrical structure can further increase the numerical accuracy, the paper by Fan and Zeng (2009) developed a new sigma point construction strategy for the unscented Kalman filter (UKF), namely, geometric simplex sigma points (GSSP). This comment presents a different perspective from the standpoint of the numerical integration. In this respect, the GSSP constitutes an integration formula of degree 2 with equal weights. Then, we demonstrate that the GSSP can be derived through the orthogonal transformation from the basic points set of degree 2. Moreover, the method presented in this comment can be used to construct more accurate sigma points set for certain dynamic problems.

Comments on "Quaternion-Based Method for SINS/SAR Integrated Navigation System"

We comment on a paper which describes a quaternion-based method for the strap-down inertial navigation systems/synthetic aperture radar (SINS/SAR) integrated navigation system. Although the paper proposes an interesting strategy, there exists a flaw in the algorithm regarding the quaternion averaging involved in the unscented particle filtering (UPF) algorithm. This letter suggests how to fix the problem.

Iterated multiplicative extended kalman filter for attitude estimation using vector observations

This paper proposes an iterated multiplicative extended Kalman filter (IMEKF) for attitude estimation using vector observations. In each iteration, the vector-measurement model is relinearized based on a new reference quaternion refined by the attitude-error estimate. An implicit reset operation on the attitude error is performed in each iteration to obtain the refined quaternion.With only a little additional computation burden, the IMEKF can much improve on the performance of the MEKF. For large initialization errors, the IMEKF performs even better than the unscented quaternion estimator but with much smaller computational burden. Numerical results are reported to validate its effectiveness and prospect in spacecraft attitude-estimation applications.