Cubature Kalman Filter Algorithm Research Articles

Dual-Mode Square Root Cubature Kalman Filter for Miniaturized Underwater Profiler Dead Reckoning

Miniaturized underwater profilers have the advantages of small size, low cost, and flexible deployment, which play together an important role in the persistent monitoring of a designated ocean area. Based on the demand for ocean observation, the system design, dynamic modeling, and dead reckoning algorithm of a miniaturized underwater profiler are carried out. The process of the persistent monitoring of designated area with a miniaturized underwater profiler is analyzed, and the overall structure of the miniaturized underwater profiler was designed. Additionally, the kinematic and dynamic model of the miniaturized underwater profiler is given based on the inertial and body coordinate systems. Then, according to the state equation and the observation equation of the miniaturized underwater profiler, a dual-mode square root cubature Kalman filter (DSRCKF) algorithm, combining standard SRCKF and adaptive SRCKF, is proposed for the dead reckoning of the miniaturized underwater profiler. Finally, a comprehensive comparison of EKF, UKF, CKF, SRCKF, and DSRCKF and the associated simulations and experiments are conducted to verify the performance of the underwater dead reckoning algorithm for the miniaturized underwater profiler. The results show that the DSRCKF algorithm has a certain role in the dead reckoning of the miniaturized underwater profiler, particularly in the z direction and attitude. In future, a real underwater dead reckoning test using miniaturized underwater profiler will be carried out.

A Binocular Color Line-Scanning Stereo Vision System for Heavy Rail Surface Detection and Correction Method of Motion Distortion.

Thanks to the line-scanning camera, the measurement method based on line-scanning stereo vision has high optical accuracy, data transmission efficiency, and a wide field of vision. It is more suitable for continuous operation and high-speed transmission of industrial product detection sites. However, the one-dimensional imaging characteristics of the line-scanning camera cause motion distortion during image data acquisition, which directly affects the accuracy of detection. Effectively reducing the influence of motion distortion is the primary problem to ensure detection accuracy. To obtain the two-dimensional color image and three-dimensional contour data of the heavy rail surface at the same time, a binocular color line-scanning stereo vision system is designed to collect the heavy rail surface data combined with the bright field illumination of the symmetrical linear light source. Aiming at the image motion distortion caused by system installation error and collaborative acquisition frame rate mismatch, this paper uses the checkerboard target and two-step cubature Kalman filter algorithm to solve the nonlinear parameters in the motion distortion model, estimate the real motion, and correct the image information. The experiments show that the accuracy of the data contained in the image is improved by 57.3% after correction.

Strong tracking adaptive window multi‐innovation cubature Kalman filter algorithm for lithium‐ion battery state of energy estimation

SummaryAccurate estimation of lithium‐ion battery state of energy (SOE) is an important prerequisite for prolonging battery life and ensuring battery safety. To achieve a high‐precision estimation of the SOE, while a ternary lithium‐ion battery being the specifically targeted in this study, a novel method for SOE estimation is proposed, which combines limited‐memory recursive least squares (LM‐RLS) with strong tracking adaptive window Multi‐innovation cubature Kalman filtering (STW‐MCKF). In the LM‐RLS algorithm, the model parameters at the current time are updated with a limited dataset to solve the data saturation problem and improve the recognition accuracy of the RLS algorithm. The CKF algorithm is optimized by the STW algorithm in the STW‐CKF algorithm to enhance its robustness under strong disturbances. Additionally, a self‐adaptive window multiple innovation strategy is proposed to improve the accuracy of SOE estimation and the stability of the CKF algorithm, while maintaining a balance between computational complexity and SOE estimation accuracy. To validate the effectiveness of the algorithm, experiments are conducted under DST and BBDST conditions. The results show that the STW‐MCKF algorithm has a maximum convergence time of 4 s and an SOE estimation error within 1.04% under DST conditions. Under BBDST conditions, the maximum convergence time is 3 s, and the SOE estimation error is within 2.34%. Furthermore, the STW‐MCKF algorithm demonstrates good stability under the two conditions, indicating the effectiveness of the proposed method for lithium‐ion battery SOE estimation.

Decoupled incremental nonlinear dynamic inversion control for aircraft autonomous landing with ground-effect

Due to motion coupling and increased model dependence, the use of nonlinear controllers in real-time applications for aircraft is extremely constrained. This explains why proportional, derivative, and integral (PID) controls are frequently used in practice. However, fast changes in the commanded path and the requirement for high accuracy and tracking performance limit use of PID in landing. This paper presents a design of a decoupled incremental nonlinear dynamic inversion (INDI) controller for performing autonomous landing of a fixed-wing aircraft. The investigation is carried out using the ground effect model and is compared without it. A carrot chasing guidance algorithm is implemented for waypoint navigation to generate the command for the proposed controller's outer loop. The navigational state estimate issue has been resolved using the Cubature Kalman Filter (CKF) approach. Two decoupled CKF algorithms named complementary Cubature Kalman Filter (CCKF) and Integrated Cubature Kalman Filter (I-CKF) are proposed to estimate aircraft navigational states comprised of attitudes, navigational position in terms of latitude, longitude altitudes, and North-East-down velocities. The mathematical model for flare and approach trajectory is designed and commanded to follow the same by the proposed INDI controller. The effectiveness of INDI have been evaluated in a simulation environment by landing the aircraft in the same runway position at arbitrary wind conditions. The performance of the proposed INDI controller is juxtaposed with a conventional proportional integrator and derivative (PID) based landing controller. The simulation findings demonstrate that the INDI has a competitive edge over traditional PID approaches due to its robust performance, less dependence on model dynamics, and fast command tracking capability required during the flare phase.

An Ultra-Wideband Indoor Localization Algorithm with Improved Cubature Kalman Filtering Based on Sigmoid Function

In this paper, an improved cubature Kalman filtering (CKF) is proposed using the Sigmoid function to address the problems of positioning accuracy degradation and large deviations in ultra-wideband (UWB) indoor positioning in non-line-of-sight environments. The improved CKF is based on the squared range difference (SRD) model of the time difference of arrival (TDOA) algorithm. The inaccurate impact of model estimation under non-Gaussian noise is reduced by updating the measurement noise matrix in real time. The covariance matrix is estimated using singular value decomposition (SVD) to solve the problem of degraded state estimation performance. The filtering effect of the improved CKF algorithm is evaluated by referring to the checkpoints in the dynamic trajectory. The experimental results show that the proposed algorithm effectively mitigates the impact of UWB ranging outliers in the occluded experimental environment, which makes the dynamic positioning trajectory smoother, better fitted, and more stable. The algorithm improves the positioning accuracy by up to 39.29% compared with the SRD model used alone.

Sideslip Angle Fusion Estimation Method of Three-Axis Autonomous Vehicle Based on Composite Model and Adaptive Cubature Kalman Filter

A sideslip angle fusion estimation strategy of three-axis vehicle based on adaptive cubature Kalman filter is investigated in this paper. According to the dynamics model, kinematics model of the three-axis vehicle, and considering the influence of tire nonlinearity, the vehicle state estimators under different conditions are designed by using the adaptive cubature Kalman filter algorithm. The dynamic-model-based estimator with linear tire model, dynamic-model-based estimator with nonlinear tire model, and kinematical-model-based estimator are proposed, then, according to the application characteristics of different estimators, a fusion estimation strategy of vehicle sideslip angle based on adaptive fuzzy weight controllers is designed, so as to improve the overall estimation accuracy by integrating the advantages of the three estimators. The simulation and experimental results show that the presented fusion estimation strategy can effectively improve the estimation accuracy of vehicle sideslip angle, and the comprehensive estimation accuracy reaches 94.37%.

Accurate state of temperature estimation for Lithium-Ion batteries based on square root cubature Kalman filter

Accurate state of temperature estimation for lithium-ion batteries is essential to ensure its safe and efficient operations. In this study, a state of temperature estimation framework for cylindrical lithium-ion batteries is proposed based on square root cubature Kalman filter algorithm, wherein the internal and surface temperatures of power battery are accurately estimated. Firstly, the electro-thermal coupling characteristics of lithium-ion battery is characterized based on an equivalent circuit model, Bernardi battery heat generation model and two-state lumped thermal model. On this basis, the discrete time-domain characteristics of internal and external temperature of the battery are addressed. Then, the genetic algorithm and forgetting factor recursive least squares algorithm are respectively employed to identify the circuit parameters and thermal physical parameters, and the internal and surface temperatures prediction is achieved based on square root cubature Kalman filter algorithm. The feasibility of the proposed temperature estimation method is verified at different charge and discharge process under wide range ambient temperature of -20 °C to 60 °C. The experimental results manifest that the proposed method can accurately online estimate the internal temperature and surface temperature with the mean absolute error of less than 1.16 °C.

Fast Passive Anti-Islanding Strategy for AC Microgrids Using Cubature Kalman Filtering Algorithm

Fast Passive Anti-Islanding Strategy for AC Microgrids Using Cubature Kalman Filtering Algorithm

Road slope estimation for heavy-duty vehicles under the influence of multiple source factors in real complex road environments

Accurate road slope information is of paramount importance for optimizing vehicle control and enhancing energy efficiency. In the context of heavy-duty vehicles (HDVs), the precise estimation of road slope presents a formidable challenge. This challenge is rooted in the coupling between vehicle mass and road slope, the robustness of road slope estimation methods across diverse driving styles, and the generalizability of algorithms to real-world road scenarios. To address these challenges, this paper proposes an innovative road slope estimation scheme that comprehensively considers the influences of multiple source factors in a real-world HDV transportation scenario. Firstly, in order to solve the problem of road slope estimation error under complex road types and different driving styles, a complex road slope estimation method based on the nonlinear predictive filter (NPF) combined with the Cubature Kalman filter (CKF) is constructed. To eliminate the problem of vehicle power fluctuation in special traffic scenarios, we propose a special scenario road slope estimation method based on the combination of a low-pass filter (LPF) and the NPF_CKF algorithm. To address the error problem when using low-cost Inertial Measurement Unit (IMU) sensors, we designed a Kalman filter (KF)-based multi-sensor fusion correction method. Finally, we conduct real-world experiments on complex roads at the test site to verify the accuracy of the proposed estimation method and its insensitivity to model parameters. The robustness and applicability of the proposed method are also verified in free-driving experiments conducted in real-world road scenarios.

Soc Estimation of Li-ion Battery Based on Adaptive CKF Algorithm

A second-or der RC equivalent circuit model was established to improve the estimation accuracy of state of charge (SOC) of power Li-ion batteries, and the model parameters were identified by the recursive least square method with forgetting factor (FFRLS). On this basis, an adaptive cubature kalman filter (ACKF) algorithm was proposed to adaptively modify the process noise covariance matrix and the measurement noise covariance matrix to improve the SOC estimation accuracy. Finally, the SOC estimation algorithm was verified by MATLAB simulations. The results show that compared with UKF and CKF algorithms, the proposed algorithm has higher estimation accuracy and robustness, and can meet the application requirements.

Estimation algorithm of pod installation angles based on cubature Kalman filter

To estimate the installation angles of the pod, a detailed derivation of the attitude transformation relationship between the pod, aircraft, and inertial navigation systems is performed. Based on this, a cubature Kalman filtering algorithm is proposed to estimate the pod’s installation angles. The algorithm architecture integrates the time update part of the Kalman filter and measurement update part of the cubature Kalman filter. This algorithm has been validated through a flight test, demonstrating its effectiveness in estimating the installation angles of the master inertial navigation system relative to the aircraft wing, as well as the slave inertial navigation system relative to the pod frame. After compensating for the installation angles, the standard deviations of roll, pitch, and azimuth between the two inertial systems have significantly improved: roll angle from 0.1414° to 0.0282°, pitch angle from 0.0803° to 0.0335°, and azimuth angle from 0.2629° to 0.0490°.

Vehicle State Joint Estimation Based on Lateral Stiffness.

In this study, a vehicle state joint estimation method based on lateral stiffness was applied to estimate the running states of electric vehicles driven by rear-drive, in-wheel motors. Different from the estimation methods used in other research, the joint estimator designed in this study uses the least-squares (LS) algorithm to estimate the lateral stiffness of the front and rear axles of the vehicle, deploying the high-degree cubature Kalman filter algorithm to estimate the vehicle state. We establish a three-degree-of-freedom nonlinear vehicle model with longitudinal velocity, lateral velocity, and yaw rate, and the lateral stiffness of the front and rear axles as the principal parameters. For the low-speed running state of the vehicle, a linearized magic tire model with high fitting accuracy was used to calculate the lateral force of the entire vehicle. The LS algorithm with a forgetting factor was used to design a lateral stiffness estimator to assess the front-axle and rear-axle lateral stiffness of the entire vehicle. The generalized high-degree cubature Kalman filter (GHCKF) algorithm was used to design the vehicle state estimator and further improve the GHCKF algorithm. A vehicle state estimator, using the square root generalized high-degree cubature Kalman filter (SRGHCKF), was designed. Therefore, the joint estimator, comprising a lateral stiffness estimator and a vehicle state estimator, adopts the LS-GHCKF/SRGHCKF algorithm and enables the estimation of the lateral stiffness, the longitudinal velocity, the lateral velocity, and the yaw rate of the entire vehicle during the driving process. A double lane change and slalom simulation were performed to analyze the feasibility and accuracy of the joint estimation algorithm and verify the results of the LS-GHCKF algorithm and the LS-SRGHCKF algorithm. Further, a low-speed driving experiment was carried out for electric vehicles driven by rear in-wheel motors. The inertial navigation system (INS), the global positioning system (GPS), the real-time kinematic (RTK), and an angle sensor were used to collect real-time vehicle data. The results were compared to verify the feasibility of the joint estimator and the progressiveness of the algorithm. The experimental verification and simulation both show that the vehicle state joint estimator, designed based on the LS-GHCKF/SRGHCKF algorithm, can accurately estimate the real-time state of the vehicle. Additionally, the LS-SRGHCKF algorithm shows better effectiveness and robustness than the LS-GHCKF algorithm.

Cubature Kalman Optimizer: A Novel Metaheuristic Algorithm for Solving Numerical Optimization Problems

This study introduces a new single-agent metaheuristic algorithm, named cubature Kalman optimizer (CKO). The CKO is inspired by the estimation ability of the cubature Kalman filter (CKF). In control system, the CKF algorithm is used to estimate the true value of a hidden quantity from an observation signal that contain an uncertainty. As an optimizer, the CKO agent works as individual CKF to estimate an optimal or a near-optimal solution. The agent performs four main tasks: solution prediction, measurement prediction, and solution update phases, which are adopted from the CKF. The proposed CKO is validated on CEC 2014 test suite on 30 benchmark functions. To further validate the performance, the proposed CKO is compared with well-known algorithms, including single-agent finite impulse response optimizer (SAFIRO), single-solution simulated Kalman filter (ssSKF), simulated Kalman filter (SKF), asynchronous simulated Kalman filter (ASKF), particle swarm optimization algorithm (PSO), genetic algorithm (GA), grey wolf optimization algorithm (GWO), and black hole algorithm (BH). Friedman's test for multiple algorithm comparison with 5% of significant level shows that the CKO offers better performance than the benchmark algorithms.

Vehicle State and Road Adhesion Coefficient Joint Estimation Based on High-Order Cubature Kalman Algorithm

With regard to the rear-drive in-wheel motor vehicle, this paper studies the joint estimation method for the vehicle state and road adhesion coefficient. A nonlinear seven degrees of freedom vehicle estimation model and a tire estimation model are established. A vehicle driving state estimator and a road adhesion coefficient estimator based on the generalized high-order cubature Kalman filter (GHCKF) algorithm are designed. The vehicle state estimator combines the vehicle model and the tire model to calculate the vehicle state parameters, provides the state parameters for the road adhesion coefficient estimator, and realizes the real-time estimation of the road adhesion coefficient. The exponential fading memory adaptive algorithm is used to update the measurement noise variance, and we upgrade the GHCKF to the adaptive generalized high-order cubature Kalman filter (AGHCKF), which estimates the vehicle state and road adhesion coefficient. The typical working conditions using the double GHCKF/AGHCKF estimation algorithm were simulated and analyzed. Then, high-and low-speed driving experiments based on typical working conditions were carried out. An integrated navigation system (INS), global positioning system (GPS), and real-time kinematic positioning (RTK) were used to collect the real-time data of the vehicle, and compare them with the estimated values of the joint estimator, to verify the feasibility of the vehicle-state–road-adhesion-coefficient joint estimator. We compared a high-order GHCKF algorithm, high-order improved AGHCKF algorithm, and a cubature Kalman filter (CKF) algorithm, and the simulation and experimental results show that the joint estimator using the CKF, GHCKF, and AGHCKF algorithms can realize the real-time estimation of the vehicle state and the road adhesion coefficient. The AGHCKF algorithm shows the best effectiveness and robustness of the three algorithms.

A Tight Coupling Algorithm for Strapdown Inertial Navigation System (SINS)/Global Positioning System (GPS) Adaptive Integrated Navigation Based on Variational Bayesian

Multi-source nonlinear noise exists in the process of multi-source navigation information fusion in long-endurance positioning systems in complex environments. In such engineering applications, the classical Kalman filter (KF) and the extended Kalman filter (EKF) have the phenomena of noise instability and parameter drift, which lead to the divergence of filtering results and reductions in accuracy over long periods of time. Aiming at the above problems, this paper proposes a fusion algorithm of the variational Bayesian (VB) and the cubature Kalman filter (CKF). Firstly, the system is modeled through nonlinear filtering, and the CKF error equation is established by taking the position difference and velocity difference between SINS and GPS as observation variables. Then, to address the problem of poor self-adaptation of the CKF algorithm, the variational Bayesian adaptive estimation method is introduced into the CKF algorithm, and a measurement noise variance estimation model is introduced to the process of time and measurement updates of the CKF algorithm to finally obtain the adaptive VB–CKF algorithm. The simulation results from the experimental platform show that the proposed fusion algorithm improves the combined SINS/GPS system by about 30% in terms of attitude angle accuracy and reduces speed and position estimation errors (RMSE) by about 45%. At the same time, comprehensive experiments on multiple types of sites show that compared with the CKF algorithm, the VB–CKF algorithm improves the positioning accuracy by 10% when the GPS signal is stable and improves the accuracy by about 38% when the GPS measurement noise changes dramatically in complex terrain, which effectively suppresses the accuracy divergence of the CKF algorithm and has high value for engineering applications.

Dynamic event-based forecasting-aided state estimation for active distribution systems subject to limited communication resource

In this paper, the dynamic event-based forecasting-aided state estimation (FASE) method is developed to deal with the state estimation problem of the active distribution system (ADS) subject to communication constraints and non-linear measurements. The proposed method first constructs a state-space model of the ADS to describe system state time evolution. Secondly, to use communication resources more effectively, the dynamic event-triggered scheme (ETS) is exploited to schedule the data transmission. Aiming at the problem of the ADS in the presence of the non-linear measurement, the Gaussian integral is approximated by the spherical cubature rule to obtain the mean and covariance of the state variables after non-linear transformation. Moreover, the upper bound of the estimation error covariance containing non-triggering errors is derived, and then minimized by suitably designing filter gain, thus developing the dynamic event-triggered cubature Kalman filter (DET-CKF) algorithm to perform state estimation for ADSs. Finally, a series of simulation experiments are conducted to verify the effectiveness of the developed FASE method.

Research on the High-Speed Collision Avoidance Method of Distributed Drive Electric Vehicles

High-speed autonomous emergency collision avoidance for intelligent vehicles is a challenging task. Many existing studies have not considered the effect of the road adhesion coefficient in the vehicle high-speed collision avoidance planning layer. Additionally, in lateral control, the effect of strong coupling of vehicle motion on the system model is not considered. This effect increases the uncertainty in the collision avoidance process and correlates with driving safety. In this regard, this paper conducted relevant studies. First, a distributed drive electric vehicle (DDEV) with controlled longitudinal speed was selected as the research platform and derived the transformed square-root cubature Kalman filter (T-SCKF) algorithm with cubature point sampling rule transformation to estimate the road adhesion coefficient. Second, the road adhesion coefficient information was used to calibrate the following vehicle safety distance model and the lane change collision avoidance lateral acceleration information was used to determine the vehicle autonomous emergency collision avoidance trajectory and to optimize the planning decision of the vehicle collision avoidance process. Last, the DDEV was laterally controlled to complete the tracking of the high-speed emergency collision avoidance trajectory by the adaptive model predictive control (Adaptive MPC) method with real-time updating of the system model. The relevant validation shows that the T-SCKF algorithm proposed in this paper can effectively estimate the road adhesion coefficient. The tracking control of the high-speed collision avoidance trajectory by the adaptive MPC method has better tracking accuracy and stability than traditional MPC and linear quadratic regulator (LQR) methods.

Backward smoothing adaptive SVDCKF integrated navigation algorithm

Aiming at the problem that the Cubature Kalman filter (CKF) algorithm has a non-positive definite of the covariance matrix in GNSS/SINS integrated navigation, which leads to the failure of Cholesky decomposition and the inability to estimate GNSS measurement noise, a backward smoothing adaptive CKF integrated navigation algorithm based on Singular Value Decomposition (SVD) is proposed. Backward smoothing adaptive CKF based on Singular Value Decomposition (BS-A-SVDCKF) algorithm replaces the Cholesky decomposition in standard CKF with SVD, constructs the test threshold for backward smoothing by using the relatively stable noise of the SINS system, and adaptively estimates based on different measurement characteristics when the GNSS measurement noise changes abruptly. Experimental results show that compared with adaptive SVDCKF (A-SVDCKF), backward smoothing SVDCKF (BS-SVDCKF) and SVDCKF, the average positioning accuracy of the proposed algorithm is improved by 32.12%, 57.07% and 68.66%, the average operating efficiency is reduced by 2.19% compared with SVDCKF, and 0.57% and 3.60% higher than that of BS-SVDCKF and A-SVDCKF, which improves the positioning accuracy of the integrated navigation system and ensures real-time.

Co-estimation of state of charge and internal temperature of pouch lithium battery based on multi-parameter time-varying electrothermal coupling model

Accurate estimation of battery state of charge (SOC) and internal temperature can improve battery performance and safety. In this process, the accuracy of the battery's electrical model and thermal model and the applicability and robustness of the estimation algorithm are key. In this paper, a multi-parameter time-varying electrothermal coupling model for pouch batteries is established. The model takes into account the coupling relationship between the battery SOC and temperature changes, as well as the heat exchange between multiple surfaces of the battery and the external environment. And then through the improved hybrid pulse power characterization (HPPC) test to realize the identification of model parameters, and the changing external heat transfer conditions could be identified online. At last, an adaptive strong tracking square root cubature Kalman filter (AST-SRCKF) algorithm was designed to estimate SOC and internal temperature simultaneously. The results were verified by three dynamic conditions. The conclusion is that the AST-SRCKF has high robustness and can be quickly corrected when the initial values of SOC and temperature are unknown. And the estimation accuracy is higher than extended Kalman filter (EKF) and dual Kalman filter (DEKF). Furthermore, the simulation efficiency is improved by more than 6 % compared with EKF, and by more than 11 % compared with DEKF, which is more conducive to online real vehicle applications.

Joint estimation of SOC and SOH of lithium battery based on multi-time scale EKF-ASRCKF algorithm

To address the low accuracy of battery charged of state (SOC) estimation due to fixed model parameters and capacity, a joint multi-time scale estimation algorithm (EKF-ASRCKF) is proposed in this paper, which consists of the adaptive square root cubature Kalman filter (ASRCKF) algorithm and the extended Kalman filter (EKF) algorithm. Firstly, the model parameters and capacity on the macroscopic time scale are identified by the EKF algorithm; meanwhile, the battery capacity estimation results characterize the battery health of state (SOH). Then the ASRCKF algorithm estimates the SOC on the microscopic time scale. Finally, the experimental validation is carried out under the BBDST condition. The validation result indicates that the EKF-ASRCKF algorithm provides a good estimation of the model parameters and capacity, which further improves the estimation precision of the battery SOC.