Square-root Cubature Kalman Filter Research Articles

DVL-Aided SINS In-Motion Alignment Filter Based on a Novel Nonlinear Attitude Error Model

A novel nonlinear attitude error model based on square root cubature information filter (SR-CIF) is proposed aiming to speed up the convergence rate of DVL-aided strapdown inertial navigation system (SINS) in-motion initial alignment for autonomous underwater vehicle (AUV) when both large initial heading error and large initial level attitude errors exist The new nonlinear attitude error model and measurement model are applicable for initial alignment after a short-time coarse alignment with large attitude errors exist. The SR-CIF embeds the whole iterative process into the data structure framework of the information filtering, replacing the covariance matrix in the square root cubature Kalman filter (SR-CKF) with an information matrix which simplifies the filter initialization. The square root technology ensures the symmetry and positive definiteness of the information matrix with enhanced stability of the filter. The simulation and experimental results indicate that the proposed DVL-aided alignment filter is effective with large initial attitude errors. The rate of convergence and estimation accuracy of the SR-CIF is higher than that of the conventional SR-CKF with large attitude misalignments.

Adaptive Cubature Kalman Filter Based on the Expectation-Maximization Algorithm

A cubature Kalman filter is considered to be one of the most useful methods for nonlinear systems. However, when the statistical characteristics of noise are unknown, the estimation accuracy is degraded. Therefore, an adaptive square-root cubature Kalman filter (ASCKF) is designed to handle the unknown noise. The maximum likelihood criterion and expectation-maximization algorithm are employed to adaptively estimate the parameters of unknown noise, thus restraining the disturbance resulting from unknown noise and improving the estimation accuracy. The stability of the proposed algorithm is theoretically analyzed. Finally, simulations are carried out to illustrate that the performance of the ASCKF algorithm is much more reliable than that of a standard square-root cubature Kalman filter.

Design of Vehicle Running States-Fused Estimation Strategy Using Kalman Filters and Tire Force Compensation Method

Accurate and reliable vehicle state estimation results are very significant to the active safety, energy optimization, and the intelligent control of vehicles. In this paper, to improve the accuracy and adaptability of vehicle running state estimation, the vehicle running states fused estimation strategy is presented for in-wheel motor drive electric vehicle using the Kalman filters and tire force compensation method. The concept of electric drive wheel model (EDWM) is developed and deduced, and then, considering that the EDWM is a nonlinear model with an unknown input, the design concept of high-order sliding mode observer is used to construct the state space equation of longitudinal force. To improve the accuracy and the reliability of vehicle state estimation, an overall estimation strategy with information fusion and tire force compensation is designed, in which a weighted square-root cubature Kalman filter with an adaptive covariance matrix of measurement noise is developed for observer design. Finally, the simulations in CarSim-Simulink co-simulation model and experiments are carried out, and the effectiveness of the designed estimation strategy is validated.

Accurate Real-Time Estimation of Power System Transients Using Constrained Symmetric Strong Tracking Square-Root Cubature Kalman Filter

The paper presents a fast and accurate algorithm for estimating four significant parameters (i.e., amplitude, frequency, phase angle, and damping factor) of a typical transient signal. The method can be connoted as the constrained symmetric strong tracking square-root cubature Kalman filter (CSSTSCKF). The important aspects of the proposed algorithm are: 1) constraints are imposed on the state vectors by way of a logarithmic barrier function that is either ignored or handled heuristically; 2) symmetric sub-optimal multiple fading factors (FFs) are augmented into the predicted covariance matrix to capture sudden changes and to tune the gain matrix in real-time; moreover, symmetry of the covariance matrix is guaranteed by the influence of Cholesky triangular decomposition; 3) effect of noise can be adjusted by tuning the soften factor. Several case studies have been simulated to evaluate the proposed algorithm with respect to some of the well-known state-of-the-art methods. The real-time performance has been evaluated by flashing the filter codes into an ARM Cortex-M7 processor board and tracking the real-time signal from the experimental test bench. The results, presented herein, indicate that the CSSTSCKF remarkably outperforms all other considered techniques. Furthermore, the stability analysis of the nonlinear filter has been proved based on the constructor expression considering the boundedness of the estimation errors and other sub-items.

Joint beam and waveform selection for the MIMO radar target tracking

Abstract The simultaneous orthogonal multi-beam transmitted by a collocated multiple-input multiple-output (MIMO) radar is very effective in multi-target tracking. Since this working mode can meet the requirement of low possibility of interception; meanwhile, gain higher sensitivity and resolution. Aiming at the resource management in such a scenario, a joint beam, power and waveform selection strategy is put forward. The optimization model is established subject to a certain number of beams, power budget and waveform parameters. The criterion, which is predicted by the Kalman recursive equation, is minimizing the posterior estimate errors of multiple targets in worst cases simultaneously. Thereby, the suitable number of utilized beams, the allocated power as well as the waveform parameter in each beam for the current time epoch can be adaptively turned according to the estimate errors from the previous time epoch. We then fully demonstrate such a non-convex problem can be transformed into several convex problems. As such, the solution can be provided efficiently to meet the real-time demand. Furthermore, through obtaining the posterior estimate error by the square-root cubature Kalman filter, prediction and then adapts the three resources for next time epoch, a closed loop feedback allocation scheme is established. The simulation results show that the proposed algorithm can significantly improve the tracking performance compared with the uniform and random resource allocation strategies.

A tracking algorithm of hypersonic target based on dynamic model and square-root cubature Kalman filter

Because of the complexities of target motion features, the tracking of hypersonic target in near space is difficult. In this paper, a new adaptive tracking algorithm based on dynamic model and improved square-root cubature Kalman filter(SRCKF) is proposed. According to the target movement characteristics, the improved adaptive piecewise constant Jerk model is proposed based on the dynamic model. The improved square-root cubature Kalman filter which the model mismatch detection function is introduced to reduce the state estimation error. The simulation results indicate that the proposed method which combines the model and the estimation algorithm has higher accuracy and stability on maneuvering target tracking than other existing methods. The research provides a feasible solution to the further improvement of the real time tracking accuracy of near-space hypersonic target.

Research on Landmarks of SLAM Based on Square Root Cubature Kalman Filter

Under the large noise, the system's observability is weak, which leads to the instability of the filtering algorithm and the slow convergence speed. The algorithm of simultaneous localization and mapping based on the square-root Cubature Kalman filter (SRCKF) with Spherical Simplex(SS) was proposed in this paper. It not only abandoned the intercept term of traditional filtering algorithm, but also avoided the large amount of computation in sigma point algorithm. The new algorithm was used in the navigation, and the simulation experiment results and data showed that the number of sampling points and the amount of calculation were reduced because of the hypersphere distribution sampling. At the same time, the square root decomposition guaranteed the non-negative of the matrix, and the algorithm had s a good stability and a certain navigation precision.

Strong tracking SCKF based on adaptive CS model for manoeuvring aircraft tracking

A novel tracking algorithm is proposed by the integration of the adaptive current statistical (CS) model and the modified strong tracking (ST) square-root cubature Kalman filter (SCKF) for the manoeuvring aircraft tracking problem. Firstly, the acceleration recursion equation and the acceleration mean input estimation are combined in order to realise the adaptive adjustment of the CS model. Then, the introduced position of the fading factor is relocated from the orthogonality principle and a new formula is put forward. Additionally, the strong manoeuver detection function is established to adjust the manoeuvring frequency of the CS model. The simulation results show that the proposed algorithm possesses better tracking accuracy than the multiple-fading-factor SCKF based on the CS model, the SCKF-ST filter based on the modified CS model and the interacting-multiple-model (IMM)-SCKF. Moreover, the proposed algorithm decreases the runtime by 40% compared with the IMM-SCKF.

Fusion Algorithm Design Based on Adaptive SCKF and Integral Correction for Side-Slip Angle Observation

Vehicle side-slip angle is crucial for various vehicle active safety applications, but measuring it directly needs expensive measurement instruments and the vehicle nonlinear dynamics, parameters uncertainty, and sensor noise cause difficulties in its observation. Therefore, the accurate, affordable side-slip angle estimator is essential. Thus, a novel adaptive square-root cubature Kalman filter (ASCKF)-based estimator with the integral correction fusion is proposed. First, the square-root cubature Kalman filter (SCKF) parameters can be adjusted adaptively based on the vehicle dynamics states. Then, considering the unknown colored noises of sensors, the integral estimation is corrected by the damping item and zero-point-reset method and the integral value can compensate the estimation error caused by the vehicle nonlinear dynamics. Therefore, the accurate side-slip angle can be estimated by the adaptive fusion of the estimation and integral values. The simulation results and real-vehicle tests show that the proposed ASCKF-based fusion algorithm has better performance than both the traditional SCKF and ASCKF.

A deconvolution scheme for the stochastic metabolic/hemodynamic model (sMHM) based on the square root cubature Kalman filter and maximum likelihood estimation

Abstract Based on clinical data collected using different brain imaging and recording techniques, brain researchers built mathematical models of the activity in the human brain. To test these models they simulate them by performing on those models a virtual brain experiment, and compare the outputs from those with the real brain activity recordings. The models can be a basis for understanding what goes wrong in brain diseases and brain disorders and potentially help to create new drugs for these conditions. Metabolic Hemodynamic Model (MHM) is one of these models that describes the changes in metabolic and hemodynamic responses during functional brain activity, formulated in a continuous-discrete state space form. MHM calibration is a decisive step for successfully capture the changes in the latent variables that can not be directly observed and predicting the brain activity related to these changes, this requires having suitable techniques that permit us to estimate both the hidden states and parameters of the MHM. The method proposed in this paper is a combination of the Square Root Cubature Kalman Filter (SCKF) and Maximum Likelihood Estimation (MLE), it uses gradient-based optimization algorithms for optimizing the objective function. Numerical results obtained with simulated data are presented to illustrate the effectiveness of the proposed method to estimate the states, parameters and regenerating the BOLD signal even when the data are contaminated with high noise level. In the proposed method, it will be explained how the gradient can be calculated with a new developed SCKF-like recursion and the result, whenever there is a vast amount of data, so much less time can be spent analyzing it compared to the time spent when the data is analyzed using finite differences. The goal of these attempts is to construct a formal system that will produce theoretical results that are corresponding to what is found in reality.

Maximum correntropy square-root cubature Kalman filter with application to SINS/GPS integrated systems

For a nonlinear system, the cubature Kalman filter (CKF) and its square-root version are useful methods to solve the state estimation problems, and both can obtain good performance in Gaussian noises. However, their performances often degrade significantly in the face of non-Gaussian noises, particularly when the measurements are contaminated by some heavy-tailed impulsive noises. By utilizing the maximum correntropy criterion (MCC) to improve the robust performance instead of traditional minimum mean square error (MMSE) criterion, a new square-root nonlinear filter is proposed in this study, named as the maximum correntropy square-root cubature Kalman filter (MCSCKF). The new filter not only retains the advantage of square-root cubature Kalman filter (SCKF), but also exhibits robust performance against heavy-tailed non-Gaussian noises. A judgment condition that avoids numerical problem is also given. The results of two illustrative examples, especially the SINS/GPS integrated systems, demonstrate the desirable performance of the proposed filter.

Adaptive Navigation Algorithm Under Abnormal Measurements in Libration-Point Mission

This paper investigates X-ray pulsar navigation for earth–moon libration-point mission. A comprehensive analysis shows that abnormal measurements are inevitable and bring quite adverse effects for navigation results. To approach this problem, a new algorithm called robust square-root cubature Kalman filter is proposed. An innovation-based adaptive scale factor is introduced to adjust measurement noise covariance so that the adverse effects of abnormal measurements can be suppressed. Effectiveness of the proposed method is demonstrated in simulation results.

Target Tracking Algorithm of Automotive Radar Based on Iterated Square-root CKF

An iterated square-root cubature Kalman filter(ISRCKF) algorithm for target tracking of automotive radar is proposed in this paper, which inherits the fast and robust advantages of SRCKF, combines with Gauss-Newton iterative theory and design a algorithm to iterate update the measurement process. The filtering accuracy of target tracking algorithm of automotive radar can be further improved as the newest measurement is fully utilized. Monte-Carlo simulation experiments were carried out aim at the automotive radar target tracking problem, in which used the algorithm to compare with classical algorithms such as square-root unscented Kalman filter (SRUKF) and SRCKF. The experimental results shows that the overall filtering accuracy of this algorithm is much improved compare with other classical filtering algorithms, and the filtering accuracy can be improved with the increase of iteration number.

A New Method for State of Charge Estimation of Lithium-Ion Batteries Using Square Root Cubature Kalman Filter

State of charge (SOC) is a key parameter for lithium-ion battery management systems. The square root cubature Kalman filter (SRCKF) algorithm has been developed to estimate the SOC of batteries. SRCKF calculates 2n points that have the same weights according to cubature transform to approximate the mean of state variables. After these points are propagated by nonlinear functions, the mean and the variance of the capture can achieve third-order precision of the real values of the nonlinear functions. SRCKF directly propagates and updates the square root of the state covariance matrix in the form of Cholesky decomposition, guarantees the nonnegative quality of the covariance matrix, and avoids the divergence of the filter. Simulink models and the test bench of extended Kalman filter (EKF), Unscented Kalman filter (UKF), cubature Kalman filter (CKF) and SRCKF are built. Three experiments have been carried out to evaluate the performances of the proposed methods. The results of the comparison of accuracy, robustness, and convergence rate with EKF, UKF, CKF and SRCKF are presented. Compared with the traditional EKF, UKF and CKF algorithms, the SRCKF algorithm is found to yield better SOC estimation accuracy, higher robustness and better convergence rate.

Marine Maneuvering Target Tracking Based on Space-Based Bearing-Only Measurement

Based on monitoring and tracking the maritime maneuvering target, the passive tracking system of space-based bearing-only measurement is constructed. Due to the strong maneuverability of target and strong nonlinearity of tracking system, an algorithm based on Bayes estimation is introduced in the interacting multiple model (IMM). By deriving the equivalent description of IMM's covariance matrix in SCKF (Square-root Cubature Kalman Filter), this paper solves the problem of mismatch between the input and output of the two algorithms, and a new algorithm flowchart is proposed. The simulation results show that the proposed algorithm can adapt to the maneuverability quickly and estimate the probability of target motion model accurately. To be more accurately, the average position and speed tracking accuracy can reach 53.79m and 1.54m / s respectively, which verifies the effectiveness of the proposed algorithm.

Adaptive Strong Tracking Square-Root Cubature Kalman Filter for Maneuvering Aircraft Tracking

A novel strong tracking square-root cubature Kalman filter (SCKF) based on the adaptive current statistical (CS) model is proposed aiming at the maneuvering aircraft tracking problem. The Jerk input estimation is introduced on the basis of the modified input estimation algorithm in order to make the connection with the state process noise and the state error covariance matrix. Thus, the online-adaptive adjustment of the CS model is achieved. Additionally, the introduced position of the fading factor is re-deduced and a novel calculation method is designed in order to overcome the invalidity problem of the traditional fading factor. Two simulation scenarios are conducted to verify the effectiveness of the proposed algorithm. The simulation results show that the proposed algorithm possesses better adaptability and tracking precision than the two state-of-the-art single model filters. Moreover, the proposed algorithm decreases the runtime by 40% while maintaining the comparable performance compared with the interacting-multiple-model SCKF.

Input Forces Estimation for Nonlinear Systems by Applying a Square-Root Cubature Kalman Filter.

This work presents a novel inverse algorithm to estimate time-varying input forces in nonlinear beam systems. With the system parameters determined, the input forces can be estimated in real-time from dynamic responses, which can be used for structural health monitoring. In the process of input forces estimation, the Runge-Kutta fourth-order algorithm was employed to discretize the state equations; a square-root cubature Kalman filter (SRCKF) was employed to suppress white noise; the residual innovation sequences, a priori state estimate, gain matrix, and innovation covariance generated by SRCKF were employed to estimate the magnitude and location of input forces by using a nonlinear estimator. The nonlinear estimator was based on the least squares method. Numerical simulations of a large deflection beam and an experiment of a linear beam constrained by a nonlinear spring were employed. The results demonstrated accuracy of the nonlinear algorithm.

Nonlinear state estimation using neural-cubature Kalman filter

ABSTRACTThe cubature Kalman filter (CKF) has been widely used in solving nonlinear state estimation problems because of many advantages such as satisfactory filtering accuracy and easy implementation compared to extended Kalman filter and unscented Kalman filter. However, the performance of CKF may degrade due to the uncertainty of the nonlinear dynamic system model. To solve this problem, a neural-cubature Kalman filter (NCKF) algorithm containing a multilayer feed-forward neural network (MFNN) in CKF is proposed to further improve the estimation accuracy and enhance the robustness of CKF. In the proposed NCKF algorithm, the MFNN was used to modify the nonlinear state estimation of CKF as the measurements were processed, and the CKF was used as both a state estimator and an online training paradigm simultaneously. The experimental results show that the estimation accuracy and robustness of the proposed method are better than those of the CKF, square-root CKF and particle filter.

Model-Based Estimation of Ankle Joint Stiffness.

We address the estimation of biomechanical parameters with wearable measurement technologies. In particular, we focus on the estimation of sagittal plane ankle joint stiffness in dorsiflexion/plantar flexion. For this estimation, a novel nonlinear biomechanical model of the lower leg was formulated that is driven by electromyographic signals. The model incorporates a two-dimensional kinematic description in the sagittal plane for the calculation of muscle lever arms and torques. To reduce estimation errors due to model uncertainties, a filtering algorithm is necessary that employs segmental orientation sensor measurements. Because of the model’s inherent nonlinearities and nonsmooth dynamics, a square-root cubature Kalman filter was developed. The performance of the novel estimation approach was evaluated in silico and in an experimental procedure. The experimental study was conducted with body-worn sensors and a test-bench that was specifically designed to obtain reference angle and torque measurements for a single joint. Results show that the filter is able to reconstruct joint angle positions, velocities and torque, as well as, joint stiffness during experimental test bench movements.

Low-cost adaptive square-root cubature Kalman filter for systems with process model uncertainty

A novel low-cost adaptive square-root cubature Kalman filter (LCASCKF) is proposed to enhance the robustness of process models while only increasing the computational load slightly. It is well-known that the Kalman filter cannot handle uncertainties in a process model, such as initial state estimation errors, parameter mismatch and abrupt state changes. These uncertainties severely affect filter performance and may even provoke divergence. A strong tracking filter (STF), which utilizes a suboptimal fading factor, is an adaptive approach that is commonly adopted to solve this problem. However, if the strong tracking SCKF (STSCKF) uses the same method as the extended Kalman filter (EKF) to introduce the suboptimal fading factor, it greatly increases the computational load. To avoid this problem, a low-cost introductory method is proposed and a hypothesis testing theory is applied to detect uncertainties. The computational load analysis is performed by counting the total number of floating-point operations and it is found that the computational load of LCASCKF is close to that of SCKF. Experimental results prove that the LCASCKF performs as well as STSCKF, while the increase in computational load is much lower than STSCKF.