Square Root Unscented Kalman Filter Research Articles

Refined Aircraft Positioning Based on Stochastic Hybrid Estimation with Adaptive Square-Root Unscented Particle Filtering

The positioning of civil aviation aircraft relative to a geographic reference point on Earth in a Cartesian frame is significant to detect the deviations from the desired path, especially for high-altitude airports or special airports based on performance-based navigation (PBN). To obtain these critical deviations during aircraft approach and landing, it is fundamental to estimate the continuous flight variables and discrete flight modes simultaneously with enough accuracy. With the coordinate conversion between the North, East, and Down (NED) frame and the geographic coordinate system based on World Geodetic System 1984 (WGS-84) considered, this study proposed a non-linear stochastic hybrid estimation algorithm with adaptive square-root unscented particle filtering (ASR-UPF) to estimate the true path. The probabilities of mode transition, represented by the normal cumulative density function of continuous states, determine whether to proceed with mode transitions. In addition, the adaptive update characterized by tracking variable noise and the importance sampling distributions based on the results of square-root unscented Kalman filtering (SR-UKF), as a comparative study of continuous system filtering, were used. The experiments illustrated the ASR-UPF is able to reduce the state estimation error more effectively, and more promptly track the error caused by incorrect mode estimation with adaptability compared to the SR-UKF. A further test with real flight data indicates that the proposed method gives the refined estimation of position and azimuth in NED frame.

Attitude Calculation Method of Drilling Tools Based on Cross-Correlation Extraction and ASRUKF

As a key component of the measurement while drilling technology, the accuracy of attitude calculation is directly related to the efficiency of resource exploration. To reduce the influence of vibration, rotation, and other disturbances on the attitude sensor during drilling, a method based on cross-correlation extraction and the adaptive square-root unscented Kalman filter (ASRUKF) is proposed to solve the attitude of the drilling tool in this paper. Firstly, the error of the signal collected by the attitude sensor is compensated, and the unscented Kalman filter (UKF) is used for filtering. Then, the effective gravitational acceleration signal is extracted by the cross-correlation method. Finally, an experimental platform for simulating the fully rotating attitude measurement system is established, and the application effects of the UKF and ASRUKF in the attitude calculation are compared. Compared with the UKF, the root mean square error of the inclination angle calculated by the ASRUKF is reduced by 12.9%, and the variance is reduced by 27.3%; the root mean square error of the azimuth angle is reduced by 29.5%, and the variance is reduced by 39.9%. The experimental results show that the attitude calculation method proposed in this paper can stably and effectively improve the accuracy of the attitude calculation of drilling tools.

Open circuit voltage - state of charge curve calibration by redefining max–min bounds for lithium-ion batteries

The relationship between open circuit voltage (OCV) and state of charge (SoC) is essential for SoC estimation of lithium-ion batteries, which can be secured by either low-current OCV test or incremental OCV test, with incremental OCV test demonstrating better results. Nevertheless, low current always leads to a discharge capacity that is inconsistent with maximum available capacity, which is a key parameter in SoC estimation. Therefore, the resultant OCV-SoC curve fails to provide a consistent SoC range and thus affects SoC estimation. In this paper, OCV-SoC curves obtained from low-current OCV tests are calibrated by redefining max–min bounds to improve SoC estimation accuracy. Max–min bounds of SoC are redefined by measuring, calculating and resetting upper and lower cut-off voltages of the OCV-SoC curve. Based on second-order RC model, model parameters are identified online and adaptive square-root unscented Kalman filter (ASRUKF) algorithm is introduced to perform SoC estimation, which is proved to be precise and adaptive. The calibrated OCV-SoC curve achieves an overall better SoC estimation performance than that obtained from the low-current OCV test and incremental OCV test, which is validated by experiments of LiNiMnCoO2 (NMC) batteries at 0°C and 25°C.

Augmenting inertial motion capture with SLAM using EKF and SRUKF data fusion algorithms

This paper proposes quaternion-based extended and square-root unscented Kalman filters to estimate human body segment positions and orientations using low-cost IMUs in conjunction with a SLAM method. The Kalman filters use measurements based on SLAM output, multilink biomechanical constraints, and vertical referencing to correct errors. In addition to the sensor biases, the fusion algorithm is capable of estimating link geometries, allowing the imposition of biomechanical constraints without a priori knowledge of sensor positions. The proposed algorithms achieve up to 5.87 (cm) and 1.1 (deg) accuracy in position and attitude estimation in various scenarios of human arm movements. Compared to the EKF, the SRUKF algorithm presents a smoother and higher convergence rate but is 2.4 times more computationally demanding. After convergence, the SRUKF is up to 17% less and 36% more accurate than the EKF in position and attitude estimation, respectively.

A Novel Adaptive Square Root UKF with Forgetting Factor for the Time-Variant Parameter Identification

The unscented Kalman filter (UKF) serves as an efficient estimator widely utilized for the recursive identification of parameters. However, the UKF is not well suited for tracking time-variant parameters. Moreover, the unscented transformation (UT) used in the UKF typically relies on Cholesky decomposition to perform the square root operation of the covariance matrix. This method necessitates the matrix to maintain symmetry and positive definiteness. Due to the adverse influence of rounding error and noise, it becomes challenging to guarantee the positive definiteness of the matrix in each recursive step for practical engineering. The square root UKF (SRUKF) eliminates the need for the square root operation in the UT by directly updating the square root of the covariance matrix during each recursion. However, the SRUKF still relies on the rank 1 update to the Cholesky factorization to perform the recursive process, which also necessitates the matrix to be positive definite. Furthermore, the SRUKF is ineffective in the identification of time-variant parameters. Therefore, this paper proposes a modification to the SRUKF that ensures unconditional numerical stability by utilizing QR decomposition. Subsequently, the modified square root UKF (MSRUKF) method is enhanced by incorporating an adaptive forgetting factor that can be adjusted based on the residual information from each recursive step. This adaptation leads to the development of the adaptive SRUKF with forgetting factor (ASRUKF-FF) method, which significantly improves the tracking capability for time-variant parameters. To validate the effectiveness of the proposed method, this paper demonstrates its application in identifying the time-variant stiffness and damping parameters of a three-story frame structure. In addition, the method is employed to estimate the time-variant stiffness of the bridge excited by vehicles. The simulation results show that the proposed method has the superiority of high accuracy, strong robustness, and widespread applicability, even with incomplete measurements and inappropriate parameter settings.

A False Data Injection Attack Detection Strategy for Unbalanced Distribution Networks State Estimation

With the advance in communication facilities and information technologies, the state estimation (SE) of distribution networks is subject to intensified cybersecurity threats caused by false data injection attacks (FDIAs). To address the issues, this paper proposes a novel FDIA detection strategy for unbalanced distribution networks. The SE model and corresponding general imperfect FDIA are introduced first to emulate the attacking behavior in practice. To achieve FDIA detection, we propose a square-root unscented Kalman filter (SR-UKF) based forecasting-aided SE (FASE) to generate estimation results. By modifying the filtering step of the proposed FASE into a redundant linear regression form, random outliers can be effectively detected and suppressed by leveraging the projection statistics (PS). Afterward, based on the acquired SE results, a generalized likelihood ratio test (GLRT) is designed to detect FDIAs on consecutive snapshots. In the GLRT, the dynamic time warping (DTW) distance between two innovation sequences is set as the test variable, which is compared with the offline determined detection threshold under a specific false alarm rate. The feasibility of the proposed general imperfect FDIA and the effectiveness of the proposed FDIA detection strategy are validated through extensive numerical simulations.

State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter

Accurate state of charge (SOC) estimation is helpful for battery management systems to extend batteries’ lifespan and ensure the safety of batteries. However, due to the pseudo-positive definiteness of the covariance matrix and noise statistics error accumulation, the SOC estimation of lithium-ion batteries is usually inaccurate or even divergent using Kalman filters, such as the unscented Kalman filter (UKF) and the square-root unscented Kalman filter (SRUKF). To resolve this problem, an SOC estimation method based on the dual-coefficient tracking improved square-root unscented Kalman filter for lithium-ion batteries is developed. The method is composed of an improved square-root unscented Kalman filter (ISRUKF) and a dual-coefficient tracker. To avoid the divergence of SOC estimation due to the covariance matrix with pseudo-positive definiteness, an ISRUKF based on the QR decomposition covariance square-root matrix is presented. Moreover, the dual-coefficient tracker is designed to track and correct the state noise error of the battery, which can reduce the SOC estimation error caused by the accumulation of the battery model error using the ISRUKF. The accuracy and robustness of the SOC estimation method using the developed method are validated by the comparison with the UKF and SRUKF. The developed algorithm shows the highest SOC estimation accuracy with the SOC error within 1.5%.

Image plane wavefront sensing and Kalman filtering for automated alignment of off-axis aspherics.

Automated alignment can significantly increase optical system precision and flexibility, and reduce time and labor costs in system setup and maintenance. We present an automated alignment technique on a double off-axis parabolic mirror system, which poses challenging alignment problems due to the mirrors' high sensitivity to aberrations, rotational asymmetry, and non-orthogonality in stage adjustments. In our methodology, we use focal plane wavefront sensing to eliminate the non-common path error, increase optical throughput, and reduce cost and system complexity. We incorporate model-based optimal estimation and control to better handle the nonlinearity, model uncertainty, and noise. Using either an iterated extended Kalman filter or a square-root unscented Kalman filter as the optimal nonlinear misalignment state estimator, we are able to consistently reduce the linear misalignment from around 1mm to <5µm and the angular misalignment from around 500 to <6a r c s e c in simulation, achieving a final wavefront error of <5∗10-5 waves in the field of view when tested at wavelength 635nm. We discover a multi-state coupling effect, which implies that different misalignment states have compensating effects on system measurements, thus interfering with the estimator's observation of misalignment state changes. We further investigate the coupling's effects on alignment quality through observability analysis.

Continuous Estimation of the Lower-Limb Multi-Joint Angles Based on Muscle Synergy Theory and State-Space Model

Continuous joint angle estimation is essential for enhancing man-machine collaboration performance. However, it is challenging to estimate the complex multi-joint angle of the lower limb accurately. Firstly, a non-redundant feature extraction algorithm for muscle synergy was proposed. The non-negative matrix factorization (NMF) algorithm was used to extract the muscle activation coefficient matrix, and the muscle activation coefficient matrix was divided into non-redundant and redundant feature vectors. Then, a state-space frame model with non-redundant features as input and redundant features as measurement output to reduce system error was proposed. The square root unscented Kalman filter (SRUKF) algorithm was used to estimate the multi-joint angle of lower limbs. We recruited ten subjects to participate in seven daily activities, including going upstairs (US), downing stairs (DS), going uphill (UH), going downhill (DH), and walking at three speeds of 0.6 m/s, 1.0 m/s, and 1.4 m/s. The results showed that the average RMSE of the proposed approach for estimating hip and knee joint angles was 0.44±0.1 and 0.73±0.5, respectively, which was significantly smaller than the common neural networks (p<0.05). Particularly, the anti-interference performance of the proposed model was tested. Meanwhile, the adaptability test was carried out through the developed lower-limb multi-joint angle estimation verification system, which proved that the proposed approach could provide accurate and stable estimation results by making full use of redundant features. It can improve the safety of online applications for sEMG auxiliary equipment.

An experimental comparative study of adaptive sigma-point Kalman filters: Case study of a rigid–flexible four-bar linkage mechanism and a servo-hydraulic actuator

This paper investigates the application, design, and implementation of different adaptive sigma-point Kalman filters for nonlinear dynamic systems. It is a well-known fact that the performance of nonlinear Kalman filters depends on a prior knowledge of system noise statistics. In real-time applications, this knowledge is difficult and in some cases impossible to predefine. Therefore, an adaptation mechanism based on the random weighting method is proposed in this work to estimate, simultaneously, noise covariance and the state of the system. On the other hand, for real time execution, the computational efficiency of such filters should also be considered to minimize the burden on computational resources. Based on these two factors, in this paper, four different variants of the adaptive sigma-point Kalman filter are studied in detail: (I) an Adaptive Unscented Kalman filter (AUKF), (II) an Adaptive Square-root Unscented Kalman filter (ASr-UKF) with an Unscented Transformation sigma-point selection strategy, (III) an Adaptive Scaled Spherical Simplex Unscented Kalman filter (AS3UKF), and (IV) an Adaptive Square-root Scaled Spherical Simplex Unscented Kalman filter (ASr-S3UKF) with scaled spherical simplex sigma-point selection strategy. The performance of these adaptive sigma-point Kalman filters is analyzed here in depth and discussed for application to a simulated rigid–flexible four-bar linkage mechanism and a practical servo-hydraulic actuator setup under different scenarios. The simulation and the experimental results indicated that, for the cases studied in this article, the filters are competitive and present advantages and disadvantages that should be dealt with according to the requirements of the problem.

Approximating a Laplacian Prior for Joint State and Model Estimation within an UKF

Low-quality models that miss relevant dynamics lead to major challenges in model-based state estimation. We address this issue by simultaneously estimating the system's states and its model inaccuracies by a square root unscented Kalman filter (SRUKF). Concretely, we augment the state with the parameter vector of a linear combination containing suitable functions that approximate the lacking dynamics. Presuming that only a few dynamical terms are relevant, the parameter vector is claimed to be sparse. In Bayesian setting, properties like sparsity are expressed by a prior distribution. One common choice for sparsity is a Laplace distribution. However, due to disadvantages of a Laplacian prior in regards to the SRUKF, the regularized horseshoe distribution, a Gaussian that approximately features sparsity, is applied instead. Results exhibit small estimation errors with model improvements detected by an automated model reduction technique.

Estimating States and Model Uncertainties Jointly by a Sparsity Promoting UKF

State estimation when only a partial model of a considered system is available remains a major challenge in many engineering fields. This work proposes a joint, square-root unscented Kalman filter to estimate states and model uncertainties simultaneously by linear combinations of physics-motivated library functions. Using a sparsity promoting approach, a selection of those linear combinations is chosen and thus an interpretable model can be extracted. Results indicate a small estimation error compared to a traditional square-root unscented Kalman filter and exhibit the enhancement of physically meaningful models.

A SLAM Algorithm Based on Edge-Cloud Collaborative Computing

Simultaneous localization and mapping (SLAM) is a typical computing-intensive task. Based on its own computing power, a mobile robot has difficult meeting the real-time performance and accuracy requirements for the SLAM process at the same time. Benefiting from the rapid growth of the network data transmission rate, cloud computing technology begins to be applied in the robotics. There is the reliability problem caused by solely relying on cloud computing. To compensate for the insufficient airborne capacity, ensure the real-time performance and reliability, and improve the accuracy, a SLAM algorithm based on edge-cloud collaborative computing is proposed. The edge estimates the mobile robot pose and the local map using a square root unscented Kalman filter (SR-UKF). The cloud estimates the mobile robot pose and the global map using a distributed square root unscented particle filter (DSR-UPF). By using sufficient particles in the cloud, DSR-UPF can improve the SLAM accuracy. The cloud returns the particle with the largest posteriori probability to the edge, and the edge performs edge-cloud data fusion based on probability. Both the simulation and the experimental results show that the proposed algorithm can improve the estimation accuracy and reduce the execution time at the same time. By transferring the heavy computation from robots to the cloud, it can enhance the environmental adaptability of mobile robots.

An Uncalibrated and Accurate Robotic Puncture Method Under Respiratory Motion

Robotic puncture system (RPS) has been widely concerned in thoracoabdominal puncture surgery, but it usually requires cumbersome hand-eye calibration (HEC) to ensure accuracy. An uncalibrated and accurate robotic puncture method under respiratory motion is proposed to determine the relationship between the manipulator and optical tracking system (OTS) in real-time. This method explored RPS to achieve the accurate location of angle and position without calibration respectively, and constructed the angle Jacobian matrix and position Jacobian matrix. An efficient approach that used Square-Root Unscented Kalman Filter (SR-UKF) to online estimate the Jacobian matrices of the RPS was presented. Moreover, this method is less time-consuming and can achieve a dynamic following, making it more efficient. Experimental studies on our designed RPS show that the proposed method has good accuracy and robustness in angle and position location; the average error of puncture results on the respiratory simulation model is 2.2382mm, which can meet the accuracy requirements of the actual thoracoabdominal puncture surgery.

Online joint estimator of key states for battery based on a new equivalent circuit model

Accurate estimation of the key states of batteries is essential for safe and reliable battery operation. In this paper, a joint estimation method for state of health (SOH), state of charge (SOC) and state of power (SOP) of batteries based on the autoregressive equivalent circuit model (AR-ECM) is proposed. Firstly, considering the coupling relationship existing between these key states of the battery, the state space-coupling model based on AR-ECM is proposed. Then, by analyzing the different characteristics of the model parameters, a differentiated model parameter identification strategy is proposed. Finally, based on the accurate estimation of the model parameters, the square root unscented Kalman filter (SR-UKF) is used to realize the online estimation of SOH and SOC. The SOP estimation under multiple constraints is realized based on the updated state, voltage, and current. Experimental results in noise free and Gaussian white noise environments show that the multi-state joint estimation algorithm has high estimation accuracy and robustness.

Adaptive state-of-charge estimation of lithium-ion batteries based on square-root unscented Kalman filter

This paper proposes a state of charge (SOC) estimation method for lithium-ion batteries. Firstly, a fractional second-order RC circuit model of the battery is established. Then, a particle swarm optimization algorithm with a linear differential decline strategy is adopted to identify the model parameters, and the accuracy of the parameterized model is verified. Finally, an adaptive fractional-order square root unscented Kalman filter (AFSR-UKF) is developed, which is able to update the noise information in real time and to overcome divergence caused by inappropriate noise covariance matrices. The effectiveness of the SOC estimation method based on the AFSR-UKF is verified under a variety of operational conditions in the perspective of the root-mean-squared error, which is shown to be below 1.0%, including at extreme temperatures, revealing good accuracy and robustness.

Adaptive Square-Root Unscented Kalman Filter Phase Unwrapping with Modified Phase Gradient Estimation

Phase unwrapping (PU) is a key program in data processing in the interferometric synthetic aperture radar (InSAR) technique, and its accuracy directly affects the quality of final SAR data products. However, PU in regions with large gradient changes and high noise has always been a difficult problem. To overcome the limitation, this article proposes an adaptive square-root unscented Kalman filter PU method. Specifically, a modified phase gradient estimation (PGE) algorithm is proposed, in which a Butterworth low-pass filter is embedded, and the PGE window can be adaptively adjusted according to phase root-mean-square errors of pixels. Furthermore, the outliers of the PGE results are detected and revised to obtain high-precision vertical and horizontal phase gradients. Finally, the unwrapped phase is calculated by the adaptive square-root unscented Kalman filter method. To the best of our knowledge, this article is the first to combine the modified PGE with an adaptive square-root unscented Kalman filter for PU. Two sets of simulated data and a set of TerraSAR-X/TanDEM-X real data were used for experimental verification. The experimental results demonstrated that the various improvement measures proposed in this article were effective. Additionally, compared with the minimum-cost flow algorithm (MCF), statistical-cost network-flow algorithm (SNAPHU) and unscented Kalman filter PU (UKFPU), the proposed method had better accuracy and model robustness.

State-of-Charge Estimation of Lithium-Ion Batteries Based on Fractional-Order Square-Root Unscented Kalman Filter

The accuracy of the state-of-charge (SOC) estimation of lithium batteries affects the battery life, driving performance, and the safety of electric vehicles. This paper presents a SOC estimation method based on the fractional-order square-root unscented Kalman filter (FSR-UKF). Firstly, a fractional second-order Resistor-Capacitance (RC) circuit model of the lithium battery is derived. The accuracy of the parameterized model is verified, revealing its superiority over integer-order standard descriptions. Then, the FSR-UKF algorithm is developed, combining the advantages of the square-root unscented Kalman filter and the fractional calculus. The effectiveness of the proposed algorithm is proven under a variety of operational conditions in the perspective of the root-mean-squared error, which is shown to be below 1.0%. In addition, several experiments illustrate the performance of the FSR-UKF.

Model-Free and Uncalibrated Eye-in-Hand Visual Servoing Approach for Concentric-Tube Robots

This article proposes a model-free and uncalibrated eye-in-hand visual servoing (EiH-VS) approach for controlling concentric-tube robots (CTRs) in minimally invasive surgery (MIS). Traditionally, a closed-loop EiH-VS controller requires an accurate robot kinematic model and hand–eye calibration. However, it is difficult to model CTR accurately when considering torsion, shear, friction, interactive force, and nonlinear constitutive behavior. In this article, we map image deviations to robot actuation variables with numerically calculated image Jacobian. The estimation of dynamic image Jacobian is based on a modified adaptive square-root unscented Kalman filter (MASR-UKF). A customized measurement matrix is constructed to describe the transformation between state and observation vectors. Moreover, an adaptive gain controller is designed to accelerate convergence. As a result, no prior knowledge of the CTR kinematic model and hand–eye calibration is needed in visual servoing tasks. Simulations and experiments have been conducted. The results validated the efficacy and efficiency of the proposed methods.

Process Monitoring of Quality-Related Variables in Wastewater Treatment Using Kalman-Elman Neural Network-Based Soft-Sensor Modeling

Proper monitoring of quality-related but hard-to-measure effluent variables in wastewater plants is imperative. Soft sensors, such as dynamic neural network, are widely used to predict and monitor these variables and then to optimize plant operations. However, the traditional training methods of dynamic neural network may lead to poor local optima and low learning rates, resulting in inaccurate estimations of parameters and deviation of predictions. This study introduces a general Kalman-Elman method to monitor the effluent qualities, such as biochemical oxygen demand (BOD), chemical oxygen demand (COD), and total nitrogen (TN). The method couples an Elman neural network with the square-root unscented Kalman filter (SR-UKF) to build a soft-sensor model. In the proposed methodology, adaptive noise estimation and weight constraining are introduced to estimate the unknown noise and constrain the parameter values. The main merits of the proposed approach include the following: First, improving the mapping accuracy of the model and overcoming the underprediction phenomena in data-driven process monitoring; second, implementing the parameter constraint and avoid large weight values; and finally, providing a new way to update the parameters online. The proposed method is verified from a dataset of the University of California database (UCI database). The obtained results show that the proposed soft-sensor model achieved better prediction performance with root mean square error (RMSE) being at least 50% better than the Elman network based on back propagation through the time algorithm (Elman-BPTT), Elman network based on momentum gradient descent algorithm (Elman-GDM), and Elman network based on Levenberg-Marquardt algorithm (Elman-LM). This method can give satisfying prediction of quality-related effluent variables with the largest correlation coefficient (R) for approximately 0.85 in output suspended solids (SS-S) and 0.95 in BOD and COD.