Robust Unscented Kalman Filter Research Articles

An Adaptive and Robust UKF Approach Based on Gaussian Process Regression-Aided Variational Bayesian

In this study, both adaptive and robust one is proposed, considering that the properties of a classic unscented Kalman filter (UKF) can be degraded severely by the outliers measured in the contamination distribution and the effects of time-varying noise. An adaptive and robust UKF approach on Gaussian process regression- assisted Variational Bayesian is proposed. The Variable Bayesian (VB) method is used to statistically approximate the time-varying noise due to the strong nonlinearity and inaccuracy of the system model. At the same time, Gaussian Process Regression (GPR) has the advantage of machine learning. The observation information trained by the GPR model performs real-time prediction of the sliding window. Based on the above two points, the output source of the measurement information is estimated to realize the robustness of the filter measurement update. When the noise is uncertain and there are outliers in the measurement information, simulation and SINS/GPS integrated navigation of actual tests were performed, respectively. The test outcomes indicate that the algorithm has better nonlinear estimation performance than the traditional method, the effectiveness of the filtering algorithm proposed is verified.

Huber-Based Robust Unscented Kalman Filter Distributed Drive Electric Vehicle State Observation

Accurate and real-time acquisition of vehicle state parameters is key to improving the performance of vehicle control systems. To improve the accuracy of state parameter estimation for distributed drive electric vehicles, an unscented Kalman filter (UKF) algorithm combined with the Huber method is proposed. In this paper, we introduce the nonlinear modified Dugoff tire model, build a nonlinear three-degrees-of-freedom time-varying parametric vehicle dynamics model, and extend the vehicle mass, the height of the center of gravity, and the yaw moment of inertia, which are significantly influenced by the driving state, into the vehicle state vector. The vehicle state parameter observer was designed using an unscented Kalman filter framework. The Huber cost function was introduced to correct the measured noise and state covariance in real-time to improve the robustness of the observer. The simulation verification of a double-lane change and straight-line driving conditions at constant speed was carried out using the Simulink/Carsim platform. The results show that observation using the Huber-based robust unscented Kalman filter (HRUKF) more realistically reflects the vehicle state in real-time, effectively suppresses the influence of abnormal error and noise, and obtains high observation accuracy.

Adaptively Robust Square-Root Cubature Kalman Filter Based on Amending

To solve the problem of decreased filtering accuracy and even filter divergence for the case that the model errors and measurement outliers exist simultaneously, an adaptively robust square-root cubature Kalman filter (SRCKF) based on amending is proposed in this paper. Based on the error analysis, the judgment criterion and the amending criterion related to the innovation are set. Then the filter could overcome the influence of the measurement outliers with the robust amendment of the measurement noise covariance matrix based on the principle of the innovation covariance matching. To further deal with model errors, the new method of the adaptive amendment of the predicted state based on the innovation is developed and combined. Finally, the proposed algorithm can balance the effect of the prior predicted value and the posterior feedback measurement value on the filtering process and reduce the state estimation error. The simulation results show that the proposed algorithm can effectively suppress the negative impact of the model errors and the measurement outliers and can obtain better estimation performance and obviously decreasing running time compared with the adaptively robust unscented Kalman filter (ARUKF) and the robust multiple fading factors cubature Kalman filter (RMCKF).

An integrity monitoring algorithm for WiFi/PDR/smartphone-integrated indoor positioning system based on unscented Kalman filter

Indoor positioning navigation technologies have developed rapidly, but little effort has been expended on integrity monitoring in Pedestrian Dead Reckoning (PDR) and WiFi indoor positioning navigation systems. PDR accuracy will drift over time. Meanwhile, WiFi positioning accuracy decreases in complex indoor environments due to severe multipath propagation and interference with signals when people move about. In our research, we aimed to improve positioning quality with an integrity monitoring algorithm for a WiFi/PDR-integrated indoor positioning system based on the unscented Kalman filter (UKF). The integrity monitoring is divided into three phases. A test statistic based on the innovation of UKF determines whether the positioning system is abnormal. Once a positioning system abnormality is detected, a robust UKF (RUKF) is triggered to achieve higher positioning accuracy. Again, the innovation of RUKF is used to judge the outliers in observations and identify positioning system faults. In the last integrity monitoring phase, users will be alerted in time to reduce the risk from positioning fault. We conducted a simulation to analyze the computational complexity of integrity monitoring. The results showed that it did not substantially increase the overall computational complexity when the number of dimensions in the state vector and observation vector in the system is small (< 20). In practice, the number of dimensions of state vector and observation vector in an indoor positioning system rarely exceeds 20. The proposed integrity monitoring algorithm was tested in two field experiments, showing that the proposed algorithm is quite robust, yielding higher positioning accuracy than the traditional method, using only UKF.

Conditioned measurement fused estimate Unscented Kalman filter for underwater target tracking using acoustic signals captured by Towed array

In the paper, an algorithm named Robust Unscented Kalman filter (R-UKF) is proposed to handle the popular ocean issue called underwater passive object/target tracking in a more efficient manner than the conventional algorithms. This R-UKF algorithm using acoustic signals supplied by a Towed array is developed when two novel Techniques named as Estimate Fusion (EF) Method and Measurement Conditioning (MC) Method are applied simultaneously to the existing Unscented kalman filter (UKF). Estimate Fusion method and Measurement Conditioning methods operate on the principles of weighted averaging of measurements in space and time respectively. EF Method contributed to the improvement by believing that the collective opinion about state estimation is much better than the individual opinions. This is accomplished by relying on Multiple sensors of TA and Multiple Intermediate estimators instead of single one. On the other hand MC method contributed to the improvements by application of the soothed measurements instead of traditional ones. The soothing is possible by weighted averaging of the current and track of past sensor data. Montecarlo simulations in Matlab(R2009a) shows that, R-UKF display better performance that its base algorithm UKF. Moreover, Optimal Performance (produce low estimation errors without demanding complex processors) of R-UKF makes it even more attractive.

A Robust Unscented Kalman Filter applied to Ultra-wideband Positioning

ABSTRACT Ultra-wideband (UWB) is well suited for indoor positioning due to its high resolution and good penetration through objects. As one of nonlinear filter algorithms, unscented Kalman filter (UKF) is widely used to estimate the position. However, UKF cannot resist the effect of outliers. The performance of the filter algorithm will be inevitably influenced. In this study, a robust UKF (RUKF) method accompanied by hypothesis test and robust estimation is proposed. Furthermore, the simulation and measurement experiments are performed to verify the effectiveness and feasibility of the proposed RUKF. Simulation experiment results are given to demonstrate that the RUKF can effectively control the influences of the outliers being treated as systematic errors and large variance random errors. When the outliers come from the thick-tailed distribution, the robust estimation does not play a role, and the RUKF does not work well. The measured experiment results show that the outliers will be generated in the non-line-of-sight environment whose impact is abnormally serious. The robust estimation can provide relatively reliable optimised residuals and control the influences of the outliers caused by gross errors. We can believe that the proposed RUKF is effective to resist the effects of outliers and improves the positioning accuracy.

Constrained Robust Unscented Kalman Filter for BDS Navigation in Dense Urban Areas

In dense urban environment, Multipath (MP) and Non-line-of-sight (NLOS) signals will degrade the performance of BeiDou Navigation Satellite System (BDS) Position, Velocity and Timing (PVT). In order to mitigate this negative impact, a Constrained Robust Unscented Kalman Filter (CRUKF) is implemented based on pseudorange/Doppler shift measurements. An equivalent weight function based on the innovation vector is constructed, which can overcome the problem of performance degrading of traditional robust methods caused by the inaccurate initial state vector. Then, the motion information, navigation direction and elevation, is included to further constrain the Robust UKF (RUKF) solution. The performance of CRUKF is analyzed using two real car tests in a dense building area, Tokyo. It is shown that there is a clear correlation between MP/NLOS errors and Position Dilution of Precision (PDOP), which seriously lows the positioning accuracy. Regarding the horizontal position, the Root Mean Square Error (RMSE) of CRUKF is 5.6 m, while those of Robust Iterative Least Square (RILS) and RUKF are 15.3 m and 6.4 m, respectively. Similar improvements are presented in vertical position and velocity and hence show the superior positioning performance of CRUKF. In addition, without sensor aiding or coupling, CRUKF can be suitable for real-time application of low-cost receiver.

Adaptive Robust Unscented Kalman Filter for AUV Acoustic Navigation.

Autonomous underwater vehicle (AUV) acoustic navigation is challenged by unknown system noise and gross errors in the acoustic observations caused by the complex marine environment. Since the classical unscented Kalman filter (UKF) algorithm cannot control the dynamic model biases and resist the influence of gross errors, an adaptive robust UKF based on the Sage-Husa filter and the robust estimation technique is proposed for AUV acoustic navigation. The proposed algorithm compensates the system noise by adopting the Sage-Husa noise estimation technique in an online manner under the condition that the system noise matrices are kept as positive or semi positive. In order to control the influence of gross errors in the acoustic observations, the equivalent gain matrix is constructed to improve the robustness of the adaptive UKF for AUV acoustic navigation based on Huber’s equivalent weight function. The effectiveness of the algorithm is verified by the simulated long baseline positioning experiment of the AUV, as well as the real marine experimental data of the ultrashort baseline positioning of an underwater towed body. The results demonstrate that the adaptive UKF can estimate the system noise through the time-varying noise estimator and avoid the problem of negative definite of the system noise variance matrix. The proposed adaptive robust UKF based on the Sage-Husa filter can further reduce the influence of gross errors while adjusting the system noise, and significantly improve the accuracy and stability of AUV acoustic navigation.

Constrained Robust Unscented Kalman Filter for Generalized Dynamic State Estimation

Due to physical or control-related limitations, some dynamic state variables are constrained, such as the regulated voltage, the exciter reference voltage, etc. In addition, there exists a set of algebraic constraints that exactly confines the evolution of the system states. However, a systematic way to fully consider all inequality and equality constraints is lacking in the existing dynamic state estimation (DSE) literature. This paper proposes a general constrained robust DSE framework that is able to deal with various equality and inequality constraints. The project operator is integrated with the pseudo-measurements formulation in a unique manner to address the inequality and equality constraints, respectively, within the unscented Kalman filter (UKF) framework, which relies on the unscented transformation and is derivative-free. To this end, the derivative-free robust UKF is extended to address all the constraints while maintaining its robustness to measurement noise and bad data. Numerical results on the IEEE 39-bus system show that the proposed method exhibits the following benefits: first, improved accuracy of the state estimates in the presence of measurement noise; second, enhanced convergence speed and ability to address weakly observed dynamic state variables; and third, enhanced capability to deal with bad data and cyber attacks.

Reliable Indoor Pseudolite Positioning Based on a Robust Estimation and Partial Ambiguity Resolution Method

The unscented Kalman filter (UKF) can effectively reduce the linearized model error and the dependence on initial coordinate values for indoor pseudolite (PL) positioning unlike the extended Kalman filter (EKF). However, PL observations are prone to various abnormalities because the indoor environment is usually complex. Standard UKF (SUKF) lacks resistance to frequent abnormal observations. This inadequacy brings difficulty in guaranteeing the accuracy and reliability of indoor PL positioning, especially for phase-based high-precision positioning. In this type of positioning, the ambiguity resolution (AR) will be difficult to achieve in the presence of abnormal observations. In this study, a robust UKF (RUKF) and partial AR (PAR) algorithm are introduced and applied in indoor PL positioning. First, the UKF is used for parameter estimation. Then, the anomaly recognition statistics and optimal ambiguity subset of PAR are constructed on the basis of the posterior residuals. The IGGIII scheme is adopted to weaken the influence of abnormal observation, and the PAR strategy is conducted in case of failure of the conventional PL-AR. The superiority of our proposed algorithm is validated using the measured indoor PL data for code-based differential PL (DPL) and phase-based real-time kinematic (RTK) positioning modes. Numerical results indicate that the positioning accuracy of RUKF-based indoor DPL is higher with a decimeter-level improvement compared that of the SUKF, especially in the presence of large gross errors. In terms of high-precision RTK positioning, RUKF can correctly identify centimeter-level anomalous observations and obtain a corresponding positioning accuracy improvement compared with the SUKF. When relatively large gross errors exist, the conventional method cannot easily realize PL-AR. By contrast, the combination of RUKF and the PAR algorithm can achieve PL-AR for the selected ambiguity subset successfully and can improve the positioning accuracy and reliability significantly. In summary, our proposed algorithm has certain resistance ability for abnormal observations. The indoor PL positioning of this algorithm outperforms that of the conventional method. Thus, the algorithm has some practical application value, especially for kinematic positioning.

Robust integrated orbit and attitude estimation using geophysical data

Geophysical information such as the Earth geomagnetic field and gravity gradient (GG) data can provide a basis for autonomous concurrent orbit and attitude estimation (COAE) of satellites in low earth orbits (LEO), as magnetometers and gravity gradiometer measurements are in general functions of time, position as well as the vehicle's orientation. While gradiometer has recently been investigated just for orbit estimation (OE), the current study is focused on COAE via only utility of the GG data. To this aim, observability conditions are analyzed, where the sensitivity of the proposed COAE approach with respect to various system and roto-translational elements is also examined. Considering the nonlinear nature of the COAE problem, a modified robust unscented Kalman filter is adopted for state estimation that is enhanced by innovation-based fading factors to be robust against the modeling errors. Subsequently, a centralized fusion of GG and three-axis magnetometer (TAM) measurement data is employed to improve the accuracy and reliability of the proposed approach for space navigation. The results obtained from numerical test cases confirm feasibility and effectiveness of the proposed technique for complete COAE of space navigation in LEO satellites.

A robust unscented Kalman filter and its application in estimating dynamic positioning ship motion states

Estimating dynamic positioning ship motion states is complex if the measured nonlinear motion data have outlying data caused by faulty sensors or ocean environmental noises. To overcome the adverse effects of sensor outliers, we developed a modified unscented Kalman filter (MUKF) algorithm. An outlier detection function was first established to spot the outliers in the measurement and then embedded into the regular unscented Kalman filter (UKF) algorithm to modify the covariance of measurement noise for obtaining smooth changes of the filter gain. To verify the developed MUKF algorithm, two dynamic positioning ship motions were simulated for estimating ship motion states in the presence of measurement outliers. Four outlier scenarios with different extents of sensor faults, different points at which the outlier occurred, and outlier duration during the ship motion course were simulated. The estimated values were compared with the theoretical ones. Additional parameter sensitivity was then performed to verify the stability and convergence performance of the developed MUKF algorithm. The results estimated by the robust MUKF were accurate and reliable, regardless of the outlier scenario, indicating the robustness of the MUKF algorithm to reduce the influence of outliers on the estimation of dynamic positioning ship motion states. The implications of this study are also discussed and presented.

Robust M–M unscented Kalman filtering for GPS/IMU navigation

In this paper, a robust unscented Kalman filter (UKF) based on the generalized maximum likelihood estimation (M-estimation) is proposed to improve the robustness of the integrated navigation system of Global Navigation Satellite System and Inertial Measurement Unit. The UKF is a variation of Kalman filter by which the Jacobian matrix calculation in a nonlinear system state model is not necessary. The proposed robust M–M unscented Kalman filter (RMUKF) applies the M-estimation principle to both functional model errors and measurement errors. Hence, this robust filter attenuates the influences of disturbances in the dynamic model and of measurement outliers without linearizing the nonlinear state space model. In addition, an equivalent weight matrix, composed of the bi-factor shrink elements, is proposed in order to keep the original correlation coefficients of the predicted state unchanged. Furthermore, a nonlinear error model is used as the dynamic equation to verify the performance of the proposed RMUKF with a simulation and field test. Compared with the conventional UKF, the impacts of measurement outliers and system disturbances on the state estimation are both controlled by RMUKF.

A Robust Hybrid Filtering Method for Accurate Battery Remaining Useful Life Prediction

Accurate remaining useful life (RUL) prediction under the noisy environment is a big challenge for the health management of modern industrial systems since the extraction of the accurate data structure from heavily corrupted data is difficult. In recent years, the kernel adaptive filter (KAF) has been widely adopted to solve the robust regression problem due to its low-complexity and high-approximation capability and robustness while the applications in battery RUL prediction are still few and far between. Thus, this paper is concerned with long-term RUL prediction using the KAF method. At first, concretely speaking, a robust KAF algorithm is derived based on the double-Gaussian-mixture (DGM) cost function, which is used to learn the capacity degradation mechanism from contaminated capacity data and so as to build the long-term prediction model. Second, a robust unscented Kalman filter (UKF) algorithm employing the DGM-based cost function is developed, which is then combined with the KAF-based prediction model to realize a more accurate and reliable prediction. Under the hybrid prognostic framework, the proposed UKF algorithm is applied to filter the noisy observations. When the observation data are inaccessible, the predicted data from the off-line trained KAF-based prediction model are adopted as the approximated value of the real observations for the UKF algorithm to optimize the prediction results and to provide the uncertainty representation. The experimental results reveal that the proposed method has great robustness when the measurements contain noise and large outliers, which makes it possible to get satisfactory prediction performance without preprocessing the data manually.

Robust Unscented Kalman Filtering With Measurement Error Detection for Tightly Coupled INS/GNSS Integration in Hypersonic Vehicle Navigation

Due to the high maneuverability of a hypersonic vehicle, the measurements for tightly coupled INS/GNSS (inertial navigation system/global navigation satellite system) integration system inevitably involve errors. The typical measurement errors include outliers in pseudorange observations and non-Gaussian noise distribution. This paper focuses on the nonlinear state estimation problem in hypersonic vehicle navigation. It presents a new innovation orthogonality-based robust unscented Kalman filter (IO-RUKF) to resist the disturbance of measurement errors on navigation performance. This IO-RUKF detects measurement errors by use of the hypothesis test theory. Subsequently, it introduces a defined robust factor to inflate the covariance of predicted measurement and further rescale the Kalman gain such that the measurements in error are less weighted to ensure the filtering robustness against measurement errors. The proposed IO-RUKF can not only correct the UKF sensitivity to measurement errors, but also avoids the loss of accuracy for state estimation in the absence of measurement errors. The efficacy and superiority of the proposed IO-RUKF have been verified through simulations and comparison analysis.

A New Adaptive Robust Unscented Kalman Filter for Improving the Accuracy of Target Tracking

In target tracking, the tracking process needs to constantly update the data information. However, during data acquisition and transmission of sensors, outliers may occur frequently, and the model is disturbed by non-Gaussian noise, that affects the performance of system state estimation. In this paper, a new filtering algorithm is proposed based on QR decomposition and singular value decomposition (SVD) method, namely adaptive robust unscented Kalman filter (QS-ARUKF) to suppress the interference of outliers, nonGaussian noise as well as a model error to achieve high accuracy state estimation. An adaptive filtering algorithm based on strong tracking idea is used in modifying the state equation of unscented Kalman filter (UKF), so that the algorithm can effectively improve the tracking ability of the state model. By using the robust filtering method to construct a new cost function used to modify the measurement covariance formula of the Kalman filter, the error of measurement model can be effectively suppressed. The QR decomposition is introduced to the time update and measurement update to avoid the covariance non-positive definite. We propose the SVD method to address the problem of numerical sensitivity in the filtering process. The purpose of this method is to replace the calculation of the inverse of the filter gain matrix and further improve the robustness of the algorithm. The simulation results showed that the proposed algorithm has higher accuracy and better robustness than the traditional filtering method.

Robust adaptive unscented Kalman filter and its application in initial alignment for body frame velocity aided strapdown inertial navigation system.

In the in-motion alignment of a strapdown inertial navigation system (SINS), the unscented Kalman filter (UKF) is usually used to solve non-linear problems. The measurement noise covariance R has a direct influence on the filtering results of the alignment of the SINS. The measurement noise is assumed to follow Gaussian distribution with a constant covariance R . However, these assumptions are often not realistic, neither the Gaussianity nor the constant covariance. This will degrade the performance of the UKF. To solve this problem, this paper proposes a novel adaptive robust UKF (NARUKF). In the NARUKF, a sliding window is used in estimating the covariance R in real-time. The NARUKF is divided into three main steps, the first step is to use the Mahalanobis distance algorithm to robustify the UKF. The second step is to use the projection statistics algorithm to reweight the abnormal stored innovations. Finally, the covariance R is adaptively estimated. The simulation and experimental results for the problem of the body frame velocity aided SINS in-motion alignment under heavier-tail distribution and/or outlier conditions demonstrate the superiority of the proposed method over the traditional ones.

M-estimator based robust unscented Kalman filter through statistical linearization

The robust unscented Kalman filter (UKF) is revisited in this paper from a new point of view, namely the statistical linear regression (SLR) perspective of the unscented transformation (UT). When the actual distribution of the observation noise deviates from the assumed Gaussian distribution, the performance of the filter may degrade significantly owing to lacking robustness. After linearizing the nonlinear observation equation using the SLR perspective, the M-estimator is employed to replace the weighted least-squares one, resulting in the robust UKF. Besides providing new theoretical aspects, the advantageous performance of the proposed filter is re-emphasized in terms of the following three: (1) robust, the proposed filter is robust against deviations from the assumed Gaussian distribution; (2) efficient, by elaborately selecting the tuning parameter of the M-estimator, the robust filter will have a slight efficiency loss compared with the UKF when the Gaussian assumption does hold; (3) accurate, the proposed filter achieves robustness without compromising the accuracy of the UT.

Power System Robust Decentralized Dynamic State Estimation Based on Multiple Hypothesis Testing

This paper proposes a fast and robust unscented Kalman filter based decentralized dynamic state estimator (DSE) for power system online monitoring and control. The proposed robust DSE is able to detect, identify, and suppress three types of outliers, namely the observation, innovation, and structural outliers. Observation outliers refer to the received PMU measurements providing unreliable metered values due to gross errors or cyber attacks; innovation outliers are typically caused by impulsive system process noise, whereas structural outliers are induced by incorrect parameters of the generators or its associated controllers, such as exciters and speed governors. To enable the fast estimation of generator states of large-scale power systems in a decentralized manner, two model decoupling approaches are presented and compared. It is shown that the generator decoupling approach presented in this paper achieves higher statistical efficiency than the ones proposed in the literature in the presence of both small and large measurement noise. To detect and distinguish three types of outliers, projection statistics based multiple hypothesis testing approach is proposed. Specifically, three hypotheses corresponding to the occurrence of three types of outliers are assumed by constructing three innovation matrices; these matrices are made up by time-correlated innovation vectors, and/or predicted states, and/or measurements; then projection statistics are applied to each of the innovation matrix and its calculated projection values are checked by a statistical test to validate the assumed hypothesis. The identified outliers are further suppressed by a generalized maximum-likelihood-type estimator. Numerical results carried out on the IEEE 39-bus system demonstrate the effectiveness and robustness of the proposed method.

Robust Frequency Divider for Power System Online Monitoring and Control

Accurate local bus frequency is essential for power system frequency regulation provided by distributed energy sources, flexible loads, and among others. This paper proposes a robust frequency divider (RFD) for online bus frequency estimation. Our RFD is independent of the load models, and the knowledge of swing equation parameters, transmission line parameters, and local Phasor Measurement Unit (PMU) measurements at each generator terminal bus is sufficient. In addition, it is able to handle several types of data quality issues, such as measurement noise, gross measurement errors, cyberattacks, and measurement losses. Furthermore, the proposed RFD contains the decentralized estimation of local generator rotor speeds and the centralized bus frequency estimation, which resembles the structure of the decentralized/hierarchical control scheme. This enables RFD for very large-scale system applications. Specifically, we decouple each generator from the rest of the system by treating metered real power injection as inputs and the frequency measurements provided by PMU as outputs; then a robust unscented Kalman filter based dynamic state estimator is proposed for local generator rotor speed estimation; finally, these rotor speeds are transmitted to control center for bus frequency estimation. Numerical results carried out on the IEEE 39-bus and 145-bus systems demonstrate the effectiveness and robustness of the proposed method.