Square-root Cubature Kalman Filter Research Articles

Automated Vehicle Sideslip Angle Estimation Considering Signal Measurement Characteristic

Vehicle slip angle (VSA) estimation is of paramount importance for connected automated vehicle dynamic control, especially in critical lateral driving scenarios. In this paper, a novel kinematic-model-based VSA estimation method is proposed by fusing information from a global navigation satellite system (GNSS) and an inertial measurement unit (IMU). First, to reject the gravity components induced by the vehicle roll and pitch, a vehicle attitude angle observer based on the square-root cubature Kalman filter (SCKF) is designed to estimate the roll and pitch. A novel feedback mechanism based on the vehicle intrinsic information (the steering angle and wheel speed) for the pitch and roll is designed. Then, the integration of the reverse smoothing and grey prediction is adopted to compensate for the cumulative velocity errors during the relatively low sampling interval of the GNSS. Moreover, the GNSS signal delay has been addressed by an estimation-prediction integrated framework. Finally, the results confirm that the proposed method can estimate the VSA under both the slalom and double lane change (DLC) scenarios.

An anti-outliers SRCKF algorithm for fixed single observer passive location

Outliers often appear in measurement noise of single observer passive location system, which has a negative effect on filtering accuracy and stability, and may even lead to filter divergence. Aiming at the effect of outliers, an anti-outliers square-root cubature Kalman filter algorithm is proposed based on Bayes theorem by using the normalized polluted normal model. In this method, the cubature rule is used to calculate the mean value and variance of the nonlinear function, the normalized contaminated normal model is used to deal with the measurement error, and the variance matrix of the measurement prediction residual is adjusted in real time according to the posterior probability of outliers. The simulation results in the fixed single observer passive location model show that the proposed algorithm is robust and can eliminate the adverse effects of discrete or continuous outliers in the measurement noise.

Multi-Target Tracking Algorithm Based on 2-D Velocity Measurements Using Dual-Frequency Interferometric Radar

Multi-target tracking (MTT) generally requires either a network of Doppler radar receivers distributed at different locations or a phased array radar. The targets moving with small/no radial velocity or angular velocity only cannot be detected and localized completely by deploying Doppler radar without antenna arrays or multiple receivers. To resolve this issue, we present a new MTT algorithm based on 2-D velocity measurements, namely, radial and angular velocities, using dual-frequency interferometric radar. The contributions of the proposed research are twofold: First, we introduce the mathematical model and implementation of the proposed algorithm by explicitly establishing the relationship between 2-D velocity measurements and kinematic state of the target in terms of Cartesian coordinates. Based on 2-D velocity measurement function, the proposed MTT algorithm comprises the following steps: (i) data association using global nearest neighbor (GNN) method (ii) target state estimation using interacting multiple model (IMM) estimator combined with square-root cubature Kalman filter (SCKF) (iii) track management using rule-based M/N logic. Second, performance of the proposed algorithm is evaluated in terms of tracking accuracy, computational complexity and IMM mean model probabilities. Simulation results for different scenarios with multiple targets moving in different tracks have been presented to verify the effectiveness of the proposed algorithm.

Multi-rate strong tracking square-root cubature Kalman filter for MEMS-INS/GPS/polarization compass integrated navigation system

The microelectromechanical system-inertial navigation system (MEMS-INS) is one of the most widely used sensor systems in navigation but must be used with other sensors because of its error accumulation characteristics. The information fusion method for integrated navigation systems based on filtering technology is thus very important. This paper introduces a new adaptive Kalman filter for nonlinear integrated systems. An improved multi-rate strong tracking square-root cubature Kalman filter (MR-STSCKF) for a MEMS-INS/Global Positioning System (GPS)/polarization compass integrated navigation system is proposed. The proposed filter is used to estimate the system covariance adaptively. The proposed approach can overcome the problem of the inconsistency between the sampling frequencies of different sensors while maintaining the high precision of the integrated navigation results. Experimental results demonstrate that the proposed MR-STSCKF algorithm is effective in improving the accuracy of the MEMS-INS/GPS/polarization compass integrated navigation system with a high sampling frequency.

Correction adaptive square-root cubature Kalman filter with application to autonomous vehicle target tracking

For a state estimation problem of nonlinear system, the square-root cubature Kalman filter (SCKF) is an effective method when the noise statistical characteristics are known. However, the performance of SCKF often degrade significantly in the face of uncertain noises interference, particularly in case of measurement or system failure. In this paper, we focus on improving the accuracy and robustness of SCKF under irregular noise. First, a weighted adaptive SCKF (WASCKF) algorithm is presented with moving window method. The WASCKF can improve the accuracy of SCKF by adaptively adjusting the covariances of measurement noise and process noise. Next, in order to further improve the robustness of WASCKF against the abrupt abnormal noise, a correction adaptive SCKF (CASCKF) algorithm based on fault detection mechanism is proposed. The CASCKF algorithm can detect whether there is a fault according to a statistical function of Chi-square distribution, and can judge and carry out the necessary correction processing by using an isolate rule. Finally, the performance of CASCKF is verified by numerical experiments of autonomous vehicle target tracking problem. The results show that the proposed CASCKF algorithm has good accuracy and robustness even with sudden abnormal noise interference.

UWB/PDR integrated indoor positioning method based on robust adaptive square root cubature Kalman filter

Aiming at the problem that the ultra wide band (UWB) wireless signal is easy to be interfered, a robust adaptive square root cubature Kalman filter (RA-SRCKF) is proposed to be used in UWB/PDR integrated indoor positioning. RA-SRCKF is based on square root cubature Kalman filter (SRCKF) and its statistical characteristic of measurement noise is updated with the method of first estimating and then modifying. The experimental results show that the proposed RA-SRCKF can effectively improve the accuracy and stability of UWB/PDR integrated positioning.

An estimation scheme of road friction coefficient based on novel tyre model and improved SCKF

In order to identify the road friction coefficient of the left and right sides of the vehicle more accurately, this paper presents an estimation framework based on a novel tyre model and modified square-root cubature Kalman filter (SCKF). To begin with, a novel tyre model is proposed by adaptively calculating the longitudinal and lateral stiffness and the effective friction coefficient. The tyre forces calculated by this model are more accurate than those calculated by the Brush model. Then, to avoid the influence of abnormal measurement noise on the estimation effect, we develop an improved SCKF (ISCKF) algorithm based on the maximum correntropy criterion. The algorithm can update the measurement noise covariance adaptively. Furthermore, a real-time estimation scheme of road friction coefficient is designed by combining the vehicle dynamics model with the proposed novel tyre model and ISCKF algorithm. Finally, the performance of the presented method is verified by the co-simulation of CarSim and MATLAB/Simulink. The results show that the designed estimation scheme not only has excellent robustness in the case of abnormal measurement noise interference but also has good adaptability to the uncertainty of road friction coefficients distribution.

Robust SCKF Filtering Method for MINS/GPS In-Motion Alignment.

This paper presents a novel multiple strong tracking adaptive square-root cubature Kalman filter (MSTASCKF) based on the frame of the Sage–Husa filter, employing the multi-fading factor which could automatically adjust the Q value according to the rapidly changing noise in the flight process. This filter can estimate the system noise in real-time during the filtering process and adjust the system noise variance matrix Q so that the filtering accuracy is not significantly reduced with the noise. At the same time, the residual error in the filtering process is used as a measure of the filtering effect, and a multiple fading factor is introduced to adjust the posterior error variance matrix in the filtering process, so that the residual error is always orthogonal and the stability of the filtering is maintained. Finally, a vibration test is designed which simulates the random noise of the short-range guided weapon in flight through the shaking table and adds the noise to the present simulation trajectory for semi-physical simulation. The simulation results show that the proposed filter can significantly reduce the attitude estimation error caused by random vibration.

Multi-Channel-Based Differential Pathlength Factor Estimation for Continuous-Wave fNIRS

Functional near-infrared spectroscopy (fNIRS) in brain imaging needs to be robust to subject-wise variability. The use of a fixed differential pathlength factor (DPF) per wavelength for the entire brain will degrade the accuracy of hemodynamic responses. Since the tissue composition varies within the brain, correct DPF values should be used for various emitter-detector distances and brain regions. In this article, a DPF estimation method combining a state-space model of the modified Beer-Lambert law (mBLL), a parameter model for estimating the reduced scattering coefficients, and dual square-root cubature Kalman filters (SCKFs) is proposed. To validate the proposed method, known light intensities (six channels, two wavelengths) and reference DPFs are generated using NIRFAST (a Matlab toolbox) using a presumed paradigm, known tissue properties, a Balloon model, and a finite element head model consisting of 58,818 mesh elements. Then, the DPF values are estimated using a Jacobian matrix from the head model and the mBLL. The results show that the estimated concentration changes correlate well with the reference data. Also, the estimated DPFs showed relative errors less than 1.33% maximum and 0.75% on average. A one-tailed $t$ -test revealed that the estimated DPFs matched the reference DPFs with more than 99.9% confidence. The developed method can efficiently access the actual DPFs even if emitter-detector distances vary significantly and the tissue properties are not uniform. With the developed state-space models for dual SCKFs, real-time estimation of the DPFs from one experiment to another has become plausible.

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).

Stage of Charge Estimation of Lithium-Ion Battery Packs Based on Improved Cubature Kalman Filter With Long Short-Term Memory Model

Accurate estimation of the state of charge (SOC) of lithium-ion battery packs remains challenging due to inconsistencies among battery cells. To achieve precise SOC estimation of battery packs, first, a long short-term memory (LSTM) recurrent neural network (RNN)-based model is constructed to characterize the battery electrical performance, and a rolling learning method is proposed to update the model parameters for improving the model accuracy. Then, an improved square root-cubature Kalman filter (SRCKF) is designed together with the multi-innovation technique to estimate the battery cell's SOC. Next, to cope with inconsistencies among battery cells, the SOC estimation values from the maximum and minimum cells are combined with a smoothing method to estimate the pack SOC. The robustness and accuracy of the proposed battery model and the cell SOC estimation method are verified by exerting the experimental validation under time-varying temperature conditions. Finally, real operation data are collected from an electric-scooter (ES) monitoring platform to further validate the generalization of the designed pack SOC estimation algorithm. The experimental results manifest that the SOC estimation error can be limited to 2% after convergence.

Square‐root cubature Kalman filter‐based vector tracking algorithm in GPS signal harsh environments

In a vector tracking loop (VTL) architecture, non-linearities exist in discriminator functions and pseudo-range/pseudo-range rate measurement expressions. Generally, normalisation functions are used in discriminators to export the desired code phase or carrier frequency error and the extended Kalman filter is adopted to estimate receiver's states. This process could be accurate enough when the code phase or carrier frequency error approaches zero in the signal moderate environment but begins to distort due to non-linearity when the tracking errors become large in harsh situations. This finally narrows the applicable range of VTL. To overcome this issue, a square-root cubature Kalman filter (CKF)-based VTL is designed in this study. The discriminator functions are employed directly as measurements of navigation filter, and the non-linear expressions of discriminator functions in terms of the receiver's position, velocity, and time states are derived without normalisation. Then the CKF, which is competitive in high-dimensional non-linear systems, is employed in its square-root version to estimate the position, velocity, acceleration, and time states of the receiver. Comparison trial results between traditional and proposed VTL illustrate that the proposed algorithm can not only keep a superior tracking accuracy but also improves the tracking stability of VTL in <20 dB-Hz signal harsh circumstances.

A hierarchical estimation scheme of tire-force based on random-walk SCKF for vehicle dynamics control

The estimation of vehicle dynamics states is crucial for vehicle stability control and autonomous driving, especially tire-force which governs vehicular motion. However, measuring tire-force directly needs expensive measurement instruments, moreover, vehicle non-linear characteristics, vehicle parameter uncertainties, unknown key variables, and sensor noise could cause great challenges in its observation. Therefore, this paper aims to develop an accurate, affordable tire-force estimator to tackle the above-mentioned issues. A novel hierarchical estimation scheme based on random-walk square-root cubature Kalman filter (SCKF) is proposed and it works without a complex tire model and considers vehicle parameter uncertainties. The estimator scheme contains three blocks. The first block estimates related key variables of tire force observation. The second block estimates both vertical tire-force and longitudinal tire-force. The longitudinal tire-force is observed based on effective tire radius identification, and a proportional integral differential (PID) method is derived based on the Lyapunov principle. Then, a random-walk SCKF algorithm is presented to estimates lateral tire-force. To validate the effectiveness of the proposed estimation scheme, CarSim & Matlab/Simulink joint simulation and real car tests are carried out. Results show that the proposed estimation scheme's accuracy performance and its potential as an affordable solution for the tire-force observation.

Vehicle state estimation for INS/GPS aided by sensors fusion and SCKF-based algorithm

To improve the safety and stability of land vehicles, this paper explores the estimation problem for vehicle states, including lateral velocity and attitude. First, a robust sliding mode observer is introduced to improve the adaptability for uncertain inputs, especially for the varying parameters in the vehicle dynamic model and longitudinal velocity. Furthermore, theoretical studies are performed to enhance the capability of the observer. In order to mitigate errors with the integrated navigation system, sensor drift model is primarily established based on a modified Elman neural network, so as to investigate the coupling between driving motion and errors. In addition, an extended square-root cubature Kalman filter is proposed to combine measurements from different sensors, utilizing a fusion strategy, to deal with severe driving motion and state estimation problems. Finally, simulation and field tests are carried out under a variety of maneuvers and conditions. The approach is compared with existing methods and evaluated experimentally, which indicates its effectiveness in improving the accuracy of vehicle state estimation.

A Coupling Diagnosis Method for Sensor Faults Detection, Isolation and Estimation of Gas Turbine Engines

In this paper a novel fault detection, isolation, and identification (FDI&amp;E) scheme using a coupling diagnosis method with the integration of the model-based method and unsupervised learning algorithm is proposed and developed for monitoring gas turbine sensor faults, which represents an integration of Square Root Cubature Kalman Filters (SRCKF) and an improved Density-Based Spatial Clustering of Application with Noise (DBSCAN) algorithm. A detection indicator produced by SRCKF with a specific hypothesis is used for extracting sensor fault features against process and measurement noise, as well as operating conditions. Then, an improved DBSCAN is implemented based on a voting scheme to detect and isolate the faulty sensors. Finally, a residual-based fault estimation scheme is proposed to track sensor fault evolution and help to judge the types of faults. Moreover, the observability of the model involved is analyzed to verify the stable operation of the FDI&amp;E scheme. Various experiments for single and concurrent sensor fault scenarios in a dual-spool gas turbine prototype during a whole flight mission are conducted to demonstrate the effectiveness of the proposed FDI&amp;E scheme. Moreover, comparative studies confirm the superiority of our proposed FDI&amp;E scheme than the existing methods in terms of promptness and robustness of the sensor FDI.

Machine Learning for Modeling the Singular Multi-Pantograph Equations.

In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

Robust Square-Root Cubature FastSLAM with Genetic Operators

SUMMARYAn improved FastSLAM based on the robust square-root cubature Kalman filter (RSRCKF) with partial genetic resampling is proposed in this paper. In the proposed method, RSRCKF is used to design the proposal distribution of FastSLAM and to estimate environment landmarks. The proposed method does not require a priori knowledge of the noise statistics. In addition, to increase diversity, it uses the genetic operators-based strategy to further improve the particle diversity. In fact, a partial genetic resampling operation is carried out to maintain the diversity of particles. The proposed method is compared with other methods via simulation and experimental data. It can be seen from the results that the proposed method provides significantly more accurate and robust estimation results compared with other methods even with fewer particles and unknown a priori. In addition, the consistency of the proposed method is better than that of other methods.

Underwater square-root cubature attitude estimator by use of quaternion-vector switching and geomagnetic field tensor

This paper presents a kind of attitude estimation algorithm based on quaternion-vector switching and square-root cubature Kalman filter for autonomous underwater vehicle (AUV). The filter formulation is based on geomagnetic field tensor measurement dependent on the attitude and a gyro-based model for attitude propagation. In this algorithm, switching between the quaternion and the three-component vector is done by a couple of the mathematical transformations. Quaternion is chosen as the state variable of attitude in the kinematics equation to time update, while the mean value and covariance of the quaternion are computed by the three-component vector to avoid the normalization constraint of quaternion. The square-root forms enjoy a continuous and improved numerical stability because all the resulting covariance matrices are guaranteed to stay positively semidefinite. The entire square-root cubature attitude estimation algorithm with quaternion-vector switching for the nonlinear equality constraint of quaternion is given. The numerical simulation of simultaneous swing motions in the three directions is performed to compare with the three kinds of filters and the results indicate that the proposed filter provides lower attitude estimation errors than the other two kinds of filters and a good convergence rate.

A consensus-based square-root cubature Kalman filter for manoeuvring target tracking in sensor networks

In this paper, a novel distributed tracking method is proposed for the problem of manoeuvring target tracking in sensor networks. Firstly, an adaptive adjustment tracking model is established by extended state observer (ESO) theory. Then, the consensus-based square-root cubature Kalman filter (SCKF) algorithm is proposed in order to improve the global accuracy and stability. In addition, the integrated model could reduce the influence of measurement noise. Finally, simulation is performed to verify the effectiveness of the scheme, whereby comparison results show that the estimation accuracy of the method proposed is higher than that of the traditional ESO and SCKF.

Integrated navigation of GPS/INS based on fusion of recursive maximum likelihood IMM and Square-root Cubature Kalman filter

Information fusion of the GPS/INS integrated system is always related to characteristics of the inertial system and the sensor feature, yet prior knowledge is still difficult to obtain in real applications. To deal with the uncertainty of error covariance and state noise in vehicle navigation, this paper presents a novel approach, wherein the integration of Square-root Cubature Kalman Filters (SCKF) and Interacting Multiple Model (IMM) are also introduced. In the framework of IMM, the SCKFs with different covariance are designed to reflect various vehicle dynamics. Besides, since the IMM-SCKF can switch flexibly among the filters, the transition probability matrix is computed with maximum likelihood method to adapt to different noise characteristics. The performance of the proposed algorithm is guaranteed by theoretical analyses, and a series of vehicular experiments with different maneuvers are carried out in an urban environment. The results indicate that, in comparison with the CKF and the IMM-CKF, the accuracy of velocity and attitude are increased by the proposed strategy.