Robust Kalman Filter Research Articles

Interacting multiple model adaptive robust Kalman filter for process and measurement modeling errors simultaneously

This paper proposes an effective Interactive Multiple Model Adaptive Robust Kalman Filter (IMMARKF) without time delay to handle situations where both process modeling errors and measurement modeling errors exist simultaneously. Building upon the robust Centered Error Entropy Kalman Filter (CEEKF) for outlier measurements and the Adaptive Kalman Filter (AKF) for process modeling errors, the IMMARKF method combines the Gaussian optimality of the KF, the adaptability of AKF, and the robustness of CEEKF using the interacting multiple model (IMM) principle to adapt reasonably to changing application environments, and can obtain estimation results in the absence of time delay. Target tracking simulations show that compared to existing methods, the proposed method can better adapt to non-stationary noise and application environments where process anomalies and measurement anomalies occur simultaneously.

Decimeter-Level Accuracy for Smartphone Real-Time Kinematic Positioning Implementing a Robust Kalman Filter Approach and Inertial Navigation System Infusion in Complex Urban Environments.

New smartphones provide real-time access to GNSS pseudorange, Doppler, or carrier-phase measurement data at 1 Hz. Simultaneously, they can receive corrections broadcast by GNSS reference stations to perform real-time kinematic (RTK) positioning. This study aims at the real-time positioning capabilities of smartphones using raw GNSS measurements as a conventional method and proposes an improvement to the positioning through the integration of Inertial Navigation System (INS) measurements. A U-Blox GNSS receiver, model ZED-F9R, was used as a benchmark for comparison. We propose an enhanced ambiguity resolution algorithm that integrates the traditional LAMBDA method with an adaptive thresholding mechanism based on real-time quality metrics. The RTK/INS fusion method integrates RTK and INS measurements using an extended Kalman filter (EKF), where the state vector x includes the position, velocity, orientation, and their respective biases. The innovation here is the inclusion of a real-time weighting scheme that adjusts the contribution of the RTK and INS measurements based on their current estimated accuracy. Also, we use the tightly coupled (TC) RTK/INS fusion framework. By leveraging INS data, the system can maintain accurate positioning even when the GNSS data are unreliable, allowing for the detection and exclusion of abnormal GNSS measurements. However, in complex urban areas such as Qazvin City in Iran, the fusion method achieved positioning accuracies of approximately 0.380 m and 0.415 m for the Xiaomi Mi 8 and Samsung Galaxy S21 Ultra smartphones, respectively. The subsequent detailed analysis across different urban streets emphasized the significance of choosing the right positioning method based on the environmental conditions. In most cases, RTK positioning outperformed Single-Point Positioning (SPP), offering decimeter-level precision, while the fusion method bridged the gap between the two, showcasing improved stability accuracy. The comparative performance between the Samsung Galaxy S21 Ultra and Xiaomi Mi 8 revealed minor differences, likely attributed to variations in the hardware design and software algorithms. The fusion method emerged as a valuable alternative when the RTK signals were unavailable or impractical. This demonstrates the potential of integrating RTK and INS measurements for enhanced real-time smartphone positioning, particularly in challenging urban environments.

Long-range LBL underwater acoustic navigation considering Earth curvature and Doppler effect

The accuracy of underwater acoustic navigation is significantly influenced by geometry configuration and systematic errors coming from geophysical environment. Long baseline (LBL) systems exhibit superior positioning precision than other acoustic navigation systems, primarily attributable to the better Geometric Dilution of Precision (GDOP). However, the geometry configuration tends to degrade when the underwater vehicle is far away from the seafloor beacons. Factors such as acoustic ray bending error, Doppler effect and Earth curvature may seriously affect the navigation accuracy. To address these issues, we present a robust Kalman filter with systematic error compensation for long-range LBL systems. Firstly, a reversed acoustic ray tracking method is proposed for a unique phenomenon in acoustic wave propagation. Following that, we construct an acoustic positioning model with a time bias parameter, to weaken the impact of Doppler effect on acoustic ranging. This additional parameter was estimated with the state of the user vehicle in real time. At last, an observation equation of pressure gauge considering Earth curvature is presented for depth constraint. Long-range LBL experiments conducted in the South China Sea is employed to validate the performance of the proposed methods, taking the positioning results of inverted ultra-short baseline (iUSBL) system as reference. The results show that the proposed method outperform traditional method with a decrease of about 60 % in the three-dimensional root mean square (3DRMS) of navigation errors. With mentioned three improvements, the 3DRMS of the long-range LBL system with the longest distance of 20 km from the beacon center is less than 14 m. These findings can provide theoretical basis for other long-range LBL systems.

Robust Attitude Estimation for Low-Dynamic Vehicles Based on MEMS-IMU and External Acceleration Compensation.

Attitude determination based on a micro-electro-mechanical system inertial measurement unit (MEMS-IMU) has attracted extensive attention. The non-gravitational components of the MEMS-IMU have a significant effect on the accuracy of attitude estimation. To improve the attitude estimation of low-dynamic vehicles under uneven soil conditions or vibrations, a robust Kalman filter (RKF) was developed and tested in this paper, where the noise covariance was adaptively changed to compensate for the external acceleration of the vehicle. The state model for MEMS-IMU attitude estimation was initially constructed using a simplified direction cosine matrix. Subsequently, the variance of unmodeled external acceleration was estimated online based on filtering innovations of different window lengths, where the acceleration disturbance was addressed by tradeoffs in time-delay and prescribed computation cost. The effectiveness of the RKF was validated through experiments using a three-axis turntable, an automatic vehicle, and a tractor tillage test. The turntable experiment demonstrated that the angle result of the RKF was 0.051° in terms of root mean square error (RMSE), showing improvements of 65.5% and 29.2% over a conventional KF and MTi-300, respectively. The dynamic attitude estimation of the automatic vehicle showed that the RKF achieves smoother pitch angles than the KF when the vehicle passes over speed bumps at different speeds; the RMSE of pitch was reduced from 0.875° to 0.460° and presented a similar attitude trend to the MTi-300. The tractor tillage test indicated that the RMSE of plough pitch was improved from 0.493° with the KF to 0.259° with the RKF, an enhancement of approximately 47.5%, illustrating the superiority of the RKF in suppressing the external acceleration disturbances of IMU-based attitude estimation.

Research on Leak Detection of Low-Pressure Gas Pipelines in Buildings Based on Improved Variational Mode Decomposition and Robust Kalman Filtering.

Aiming at the complex characteristics of negative pressure waves in low-pressure pipelines inside of buildings, we proposed an estimation method of pressure fluctuation trends based on the robust Kalman filter and the improved VMD, which can be used for leakage detection. The reconstructed baseline signal can accurately describe the fluctuation trend of the negative pressure wave after the pressure drop, and quantitatively express the characteristic difference between the leakage condition and the gas usage condition. The robust Kalman filter was used to estimate the pressure fluctuations. The parameters of VMD were adaptively calculated based on the WAA and discrete scale space. The trend components contained in the IMFs were separated by a reconstruction based on the Fourier series. Based on the simulation signal, the method can accurately restore the trend component contained in the complex pressure signal. Based on the actual signals, the accuracy of small leakage detection is 96.7% and the accuracy of large leakage detection is 73.3%.

Robust Kalman filters based on the sub-Gaussian [formula omitted]-stable distribution

Motivated by filtering tasks under a linear system with non-Gaussian heavy-tailed noise, various robust Kalman filters (RKFs) based on different heavy-tailed distributions have been proposed. Although the sub-Gaussian α-stable (SGαS) distribution captures heavy tails well and is applicable in various scenarios, its potential has not yet been explored for RKFs. The main hindrance is that there is no closed-form expression of its mixing density. This paper proposes a novel RKF framework, RKF-SGαS, where the process noise is assumed to be Gaussian and the heavy-tailed measurement noise is modelled by the SGαS distribution. The corresponding joint posterior distribution of the state vector and auxiliary random variables is approximated by the Variational Bayesian approach. Also, four different minimum mean square error (MMSE) estimators of the scale function are presented. The first two methods are based on the Importance Sampling (IS) and Gauss–Laguerre quadrature (GLQ), respectively. In contrast, the last two estimators combine a proposed Gamma series (GS) based method with the IS and GLQ estimators and hence are called GSIS and GSGL. Besides, the RKF-SGαS is compared with the state-of-the-art RKFs under three kinds of heavy-tailed measurement noises, and the simulation results demonstrate its estimation accuracy and efficiency.

Improving the stochastic model for code pseudorange observations from Android smartphones

In recent years, there has been increasing attention to positioning, navigation, and timing applications with smartphones. Because of frequently disrupted carrier phase observations, code observations remain critical for smartphone-based positioning. Considering a realistic stochastic model is mandatory to obtain the utmost positioning performance, this study proposes a sound stochastic approach for code observations from Android smartphones. The proposed approach includes a modified version of the SIGMA-ɛ variance model with different coefficients for each GNSS constellation and a robust Kalman filter method. First the noise characteristics of observations from the Xiaomi Mi 8 smartphone are analyzed utilizing code-minus-phase combinations to estimate the coefficients for each GNSS constellation. This includes the determination of a variance model as well as a check of the probability distribution. Finally, the proposed approach is validated in the positioning domain using single-frequency code observation-based real-time standalone positioning. The results show that more than 95% of observations follow the normal distribution when the proposed approach is applied. Compared with the conventional stochastic approach, including a C/N0-dependent model and standard Kalman filter, it improves the positioning accuracy by 45.8% in a static experiment, while its improvement is equal to 26.6% in a kinematic experiment. For the static and kinematic experiments, in 50% of the epochs, the 3D positioning errors are smaller than 3.0 m and 3.4 m for the proposed stochastic approach. The results exhibit that the stochastic properties of code observations from smartphones can be successfully represented by the proposed approach.

A novel robust adaptive Kalman filter with application to urban vehicle integrated navigation systems

This study addresses the formidable challenge presented by non-Gaussian, time-varying, heavy-tailed process noise in integrated vehicle navigation systems. While robust adaptive filtering methods exist, their efficacy is constrained, and certain algorithms exhibit suboptimal usability due to an excess of a priori parameters. We deduce an adaptive robust Kalman filter, characterizing process noise with a Student's t distribution and measurement noise adhering to a Gaussian distribution. To enhance robustness against unknown or varying noise statistics, our method simultaneously estimates the scale matrix, degrees of freedom parameters, state vector, and measurement noise covariance through the expectation–maximization (EM) algorithm and Bayesian variational inference theory. Rigorous simulations and empirical experiments validate its superior accuracy and stability, requiring minimal a priori parameter information. Vehicle-based experiments further demonstrate positioning error below 2.78 m, velocity error below 0.34 m/s, and heading error less than 1.28° after achieving a steady state. This approach exhibits significant potential for advancing the precision and reliability of vehicle navigation systems.

A method for detecting faults in integrated navigation system based on improved SPRT adaptive filtering

GNSS measurement noise faults can easily cause a drop or even divergence in the filter accuracy of integrated navigation system. Therefore, real-time fault detection and processing are necessary. With the aim of the defects of residual Chi-square test and sequential probability ratio test (SPRT) in detecting measurement noise faults, an adaptive filtering algorithm based on improved SPRT detection is proposed. On the one hand, the influence of the current time innovation on the statistics is strengthened by the method of fading weighting. On the other hand, the measurement update equation of the adaptive filter is constructed by calculating the weight factor of the statistics in real-time, which improves the accuracy and robustness of the state estimation. Simulation experiments show that the proposed algorithm is more sensitive and non-missing compared to the residual Chi-square test in the case of large value abrupt faults in the measurement noise. The proposed algorithm reduces the fault detection delay time by about 65.5% compared to SPRT in the case of slow change faults in measurement noise. In the case of slow change faults with small values, the proposed algorithm reduces the missing detection rate by 40% compared to the residual Chi-square test. In addition, the proposed algorithm is compared with the robust Kalman filter based on the residual Chi-square and Sage-Husa adaptive filtering. The results show that the proposed algorithm has higher navigation accuracy when the system has slow change faults.

Robust Kalman filter for fractional order systems with uncertain colored noise variance

For fractional order systems with colored process noise, the discretization fractional order system model is used to construct the augmented vector defined by the state vector and colored process noise vector. Based on the augmented equation of fractional order systems, the robust local Kalman filtering algorithm for fractional order systems with colored process noise is derived. The matrix weighted fusion, weighted measurement fusion and centralized fusion methods were used to fuse and estimate the state of multi-sensor fractional order system. Simulation results show the effectiveness of the proposed algorithm.

Research on indoor positioning system algorithm based on UWB technology

Ultra wideband (UWB) indoor positioning technology is attracting significant attention in the present realm of indoor wireless positioning technology due to its numerous advantages, including low energy consumption, exceptional wireless transmission rates, remarkable resolution, and powerful anti-interference capabilities.This paper applies the time of arrival (TOA) location technique, derived from the research on UWB location method, to calculate the distance between the base station and the tag using the signal flight time. By incorporating the bilateral two-way ranging algorithm, the system has been fine-tuned to lower the precision demands on the hardware.A novel positioning algorithm is introduced, utilizing both C-Taylor and adaptive robust Kalman filter, which is based on the Time of Arrival (TOA) method. This algorithm aims to address the limitations of Chan and Taylor algorithms and minimize the non-line-of-sight (NLOS) error. This algorithm has excellent theoretical research value and practical application value.

Robust Kalman filters under epistemic uncertainty for non‐Gaussian systems with multiplicative noise

AbstractThis article proposes two robust Kalman filters to solve the issue of inaccurate modeling in multiplicative noise systems due to epistemic limitations. First, we construct all conceivable state/measurement transition probability densities as an ambiguity set. This ambiguity set chooses the Wasserstein distance or the moment‐based metric as the distance metric. Besides, this set is an inequality set with a chosen tolerance, which can be seen as a non‐negative radius ball. Then, by combining the robust solution of the least favorable model in that ball with the alternating direction method of multipliers or an efficient direct solution method, we propose two robust Kalman filters based on the minimum mean square error criterion. A classical example is provided to verify the effectiveness of the proposed robust filters in comparison to existing state‐of‐the‐art filters.

An improved two-phase robust distributed Kalman filter

To enhance the efficacy of the distributed filter in mitigating heavy-tailed non-Gaussian noise and accommodating intricate environments, this study investigates the utilization of absolute and relative measurement information for the realization of multi-target cooperative positioning and proposes an improved robust distributed Kalman filter in this paper. The method is divided into two phases. Firstly, absolute measurement information undergoes processing via the standard Kalman filter, yielding the Kalman estimation value for the first phase, subsequently serving as the time update value for the second phase. Next, a robust Kalman filter based on Student’s t distribution is used to process relative measurement information for measurement updates, resulting in more accurate estimates. Simulation results demonstrate that the proposed approach effectively approximates the posterior probability distribution, leading to enhanced stability and accuracy in positioning within environments characterized by both Gaussian and heavy-tailed non-Gaussian noise.

A robust Kalman filter based on Gaussian-Exponential distribution for SINS/USBL integration navigation system

In order to meet the needs of high-precision navigation for underwater vehicles, a strapdown inertial navigation system (SINS)/ultra-short baseline (USBL) tightly coupled integrated strategy is proposed, which can avoid the errors generated in the calculation of absolute position and improve the positioning accuracy of the integrated navigation system compared with the loosely coupled strategy. In addition, due to the influence of environmental noise in the ocean, the measurement of USBL is accompanied by a large number of outliers, which leads to the decrease of the precision of the integrated navigation system based on traditional Kalman filter. Therefore, a robust filter based on Gaussian-Exponential distribution is presented for the SINS/USBL integrated navigation system in this paper. Finally, the feasibility and the superiority of the proposed algorithm is verified through simulation and field test in the Yangtze River.

Low-Earth-Orbit Satellites and Robust Theory-Augmented GPS/Inertial-Navigation-System Tight Integration for Vehicle-Borne Positioning

Positioning by means of the Global Positioning System (GPS) is a traditional and widely used method. However, its performance is affected by the user environment, such as multi-path effects and poor anti-interference abilities. Therefore, an Inertial Navigation System (INS) has been integrated with GPS to overcome the disadvantages of GPS positioning. INSs do not rely on any external system information and has strong autonomy and independence from the external environment. However, the performance of GPS/INS is visibly degraded in low-observability GPS environments (tall buildings, viaducts, underground tunnels, woods, etc.). Fortunately, with the emergence of Low-Earth-Orbit (LEO) satellites in recent years, the constellation configuration can be extended with the advantages of lower orbits, greater speeds, and richer geometric structures. LEO improves the geometric structure between users and satellites and provides many more observations. Meanwhile, a robust theory approach is applied that can restrain or remove the impact of low-accuracy observations. In this study, we applied LEO data and a robust theory approach to enhance the GPS/INS tight integration. To verify the effectiveness of this method, a set of vehicles and simulated LEO data were analyzed. The results show that robust Kalman filtering (RKF) provides a visible enhancement in the positioning accuracy of GPS/INS integration. This effectively restrains the mutation error and has a smoothing effect on the positioning results. In addition, the addition of LEO data significantly improves the positioning accuracy of a sole GPS and GPS/INS integration. The GPS/LEO/INS integration has the highest positioning accuracy, with Root-Mean-Square Errors (RMSEs) of the north, east, and vertical positions of 2.38 m, 1.94 m, and 2.49 m, respectively, which corresponds to an improvement of 30.21%, 47.43%, and 34.13% compared to sole GPS-based positioning and 8.60%, 17.24%, and 12.14% when compared to the GPS/INS mode. Simultaneously, the simulation results show that LEO and INSs can improve the positioning performance of GPS under GPS-blocked conditions.

Data-Driven Model Predictive Control for Uncalibrated Visual Servoing

This paper addresses the image-based visual servoing (IBVS) control problem with an uncalibrated camera, unknown dynamics, and constraints. A novel data-driven uncalibrated IBVS (UIBVS) strategy is proposed, incorporated with the Koopman-based model predictive control (KMPC) algorithm and the adaptive robust Kalman filter (ARKF). First, to alleviate the need for calibration of the camera’s intrinsic and extrinsic parameters, the ARKF with an adaptive factor is utilized to estimate the image Jacobian matrix online, thereby eliminating the laborious camera calibration procedures and improving robustness against camera disturbances. Then, a data-driven MPC strategy is proposed, wherein the unknown nonlinear dynamic model is learned using the Koopman operator theory, resulting in a linear Koopman prediction model. Only input–output data are used to construct the prediction model, and hence, the proposed approach is robust against model uncertainties. Furthermore, with a symmetric quadratic cost function, the proposed approach solves the quadratic programming problem online, and visibility constraints as well as joint torque constraints are taken into account. As a result, the proposed KMPC scheme can be implemented in real time, and the UIBVS performance degradation which arises from the control torque constraints can be avoided. Simulations and comparisons for a 2-DOF robotic manipulator demonstrate the feasibility of the proposed approach. Simulation results further validate that the computation time of the proposed approach is comparable to the one of kinematic-based methods.

Robust Kalman Filter for Systems With Colored Heavy-Tailed Process and Measurement Noises

In this paper, we consider the robust state estimation for a linear system with colored heavy-tailed process and measurement noises. We employ the state augmentation and measurement differencing methods to whiten the colored noise and use the Student’s t distribution to model the heavy-tailed property, which makes a new state space model with the augmented state vector. The posterior estimation of the system state, inaccurate scale matrices, and auxiliary parameters are jointly inferred with the variational Bayes method by constructing the hierarchical Gaussian forms of the prediction and likelihood probability density functions and selecting the proper prior distributions of the scale matrices and auxiliary parameters. A typical target tracking simulation is given to confirm the performance of the proposed robust Kalman filter.

Variational Bayesian-based robust adaptive filtering for GNSS/INS tightly coupled positioning in urban environments

Kalman filter (KF) is an efficient approach for state estimation in integrated positioning systems of global navigation satellite systems and inertial navigation systems (GNSS/INS). Unfortunately, GNSS/INS integrated positioning systems are susceptible to state model perturbations and measurement outliers, which can severely degrade KF performance, especially in complex urban environments. To solve this problem, a robust adaptive filtering algorithm based on variational Bayesian (VBRAKF) is presented that can be employed in GNSS/INS tightly coupled positioning systems. In VBRAKF, the robust estimation approach is first introduced into the filter, followed by modeling the predicted covariance matrix using the inverse Wishart distribution, and finally the state is estimated using Variational Bayesian. The robustness is achieved by adjusting the noise by an equivalent weight matrix to reduce gross errors in observations, while the adaptiveness is realized by the variational Bayesian estimation method for the uncertain predicted noise covariance matrix. The positioning performance of the proposed algorithm in urban environments is verified through two sets of land vehicle experiments. The experimental results demonstrate that the 3D positioning accuracy of the VBRAKF algorithm is improved by 32.1 % and 23.5 % in experiment 1 and by 37.1 % and 17.3 % in experiment 2 when compared to KF and Robust Kalman Filter (RKF). The proposed VBRAKF algorithm has better positioning performance in all schemes, can simultaneously control the influence of measurement outliers and inaccurate noise statistics and improves the accuracy of filter estimation.

Robust Recursive Least-Squares Wiener Filter for Linear Continuous-Time Uncertain Stochastic Systems

For linear continuous-time systems with uncertainties in the system and observation matrices, an original robust RLS Wiener filter is designed in this study. The robust RLS Wiener filter does not assume norm-bounded uncertainty for the system and observation matrices, in contrast to the robust Kalman filter. In the design of the robust RLS Wiener filter, the degraded signal, affected by the uncertainties in the system and observation matrices, is modeled by an autoregressive (AR) model. The system and observation matrices for the degraded signal are formulated from the relationship between the AR model of the degraded signal and the state-space model. Estimation formulas for the system and observation matrices are proposed in Section 2. The robust filtering problem is introduced based on the minimization of the mean-square value of the filtering errors for the system states. The robust filtering estimate is given as an integral transformation of the degraded observations using the impulse response function. The integral equation that an optimal impulse response function satisfies is given in Section 3. Theorem 1 presents a robust RLS Wiener filtering algorithm starting from this integral equation. The proposed robust RLS Wiener filter outperforms the existing robust Kalman filter regarding estimate accuracy, as shown by a numerical simulation example.

Application of Adaptive Robust Kalman Filter Base on MCC for SINS/GPS Integrated Navigation.

In this paper, an adaptive and robust Kalman filter algorithm based on the maximum correntropy criterion (MCC) is proposed to solve the problem of integrated navigation accuracy reduction, which is caused by the non-Gaussian noise and time-varying noise of GPS measurement in complex environment. Firstly, the Grubbs criterion was used to remove outliers, which are contained in the GPS measurement. Then, a fixed-length sliding window was used to estimate the decay factor adaptively. Based on the fixed-length sliding window method, the time-varying noises, which are considered in integrated navigation system, are addressed. Moreover, a MCC method is used to suppress the non-Gaussian noises, which are generated with external corruption. Finally, the method, which is proposed in this paper, is verified by the designed simulation and field tests. The results show that the influence of the non-Gaussian noise and time-varying noise of the GPS measurement is detected and isolated by the proposed algorithm, effectively. The navigation accuracy and stability are improved.