Classical Kalman Filter Research Articles

Kalman Filtering in Non-Gaussian Model Errors: A New Perspective [Tips & Tricks

It is well known that the optimality of the Kalman filter relies on the Gaussian distribution of process and observation model errors, which in many situations is well justified <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> – <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[3]</xref> . However, this optimality is useless in applications where the distribution assumptions of the model errors do not hold in practice. Even minor deviations from the assumed (or nominal) distribution may cause the Kalman filter’s performance to drastically degrade or completely break down. In particular, when dealing with perceptually important signals, such as speech, image, medical, campaign, and ocean engineering, measurements have confirmed the presence of non-Gaussian impulsive (heavy-tailed) and Laplace noises <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[4]</xref> . Therefore, the classical Kalman filter, which is derived under the nominal Gaussian probability model, is biased and even fails in such situations.

Joint Estimation of Multi-Scale Structural Responses and Unknown Loadings Based on Modal Kalman Filter Without Using Collocated Acceleration Observations

It has been proved that the usage of multi-type observations including global and local information for structural health monitoring (SHM) outperforms that of solo-type measurement. Kalman filter (KF) is a simple but powerful tool for the online state estimation. However, external force is required for the classic KF. Moreover, direct implementation of KF in time domain may be time consuming especially for large structures with many degrees-of-freedoms (DOFs) involved. From these points of view, the idea of modal Kalman filter (MKF) is introduced. An MKF-based approach is then proposed for joint estimation of multi-scale responses and unknown loadings with data fusion of multi-type observations. By using a projection matrix and the selected several modes, a modified observation equation in modal domain is derived. Limited multi-type measurements are fused together for online estimating multi-scale responses of structure at its critical locations. The unknown excitation is simultaneously identified by least squares estimation (LSE). The drift problem of the real time estimation of structural responses and unknown loads is avoided. The effectiveness of the proposed approach is numerically demonstrated via several examples. Results show that the proposed approach is capable of satisfactorily estimating the unmeasured multi-scale responses and identifying the unknown loadings.

Error Correction Method of Sensor Based on Fuzzy-KF for Measurement of Drilling Cuttings

In oil and gas development operation, for horizontal wells and extended reach wells, monitoring cuttings flow rate is an important method for analyzing wellbore cleanliness. However, due to complex working environment on site, the accuracy of cuttings flow rate monitoring is low. In order to solve this problem, based on the analysis of a cuttings flow rate real-time measurement method, combined with the classic Kalman filter (KF) algorithm, an adaptive KF model based on fuzzy control rules (Fuzzy-KF) is proposed, it realize real-time update of process noise <inline-formula> <tex-math notation="LaTeX">$({Q})$ </tex-math></inline-formula> and measurement noise <inline-formula> <tex-math notation="LaTeX">$({R})$ </tex-math></inline-formula> and realize real-time correction of data collected by sensors of Cuttings Flow Meters (CFM). Through the comparison of experiments, the research results show that the mean square error (MSE) of the Fuzzy-KF model is 76.2919, and the posterior error estimate be less than 0.3, and all aspects of performance are better than the KF and Sage-Husa KF models. The effects of different initial values of <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${R}$ </tex-math></inline-formula> on the experimental results were studied, the research result shows that the difference of the initial value of <inline-formula> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${R}$ </tex-math></inline-formula> will affect the data correction result, but does not change the change trend of each parameter. The Fuzzy-KF model established in this paper can effectively reduce the environmental interference to CFM, and it provides new technical support for improving the accuracy of cuttings flow rate monitoring.

Eye Tracking Algorithm Based on Multi Model Kalman Filter

One of the most important pieces of Human Machine Interface (HMI) equipment is an eye tracking system that is used for many different applications. This paper aims to present an algorithm in order to improve the efficiency of eye tracking in the image by means of a multi-model Kalman filter. In the classical Kalman filter, one model is used for estimation of the object, but in the multi-model Kalman filter, several models are used for estimating the object. The important features of the multiple-model Kalman filter are improving the efficiency and reducing its estimating errors relative to the classical Kalman filter. The proposed algorithm consists of two parts. The first step is recognizing the initial position of the eye, and Support Vector Machine (SVM) has been used in this part. In the second part, the position of the eye is predicted in the next frame by using a multi-model Kalman filter, which applies constant speed and acceleration models based on the normal human eye. Doi: 10.28991/HIJ-2022-03-01-02 Full Text: PDF

Kalman Filter for Linear Systems With Unknown Structural Parameters

This brief considers Kalman filter for linear systems with unknown structural parameters. We design a Bayesian parameter identification algorithm based on maximum likelihood (ML) criterion and expectation maximization (EM). Under the identification of structural parameter, we perform the Kalman filter to estimate the states. The Kalman filter and the parameter identification algorithm are interactive to estimate the states and the structural parameters. More specifically, first, we use EM algorithm together with Kalman smoother to estimate the structural parameters. Then, on the basis of results provided by the first step, we employ the classical Kalman filter to predict and update the states, which formulates the final proposed Kalman filter. Performance analysis and simulation results verify the presented filter.

Kalman filter–random forest‐based method of dynamic load identification for structures with interval uncertainties

It is widely acknowledged that dynamic load identification is a powerful technology for structural health monitoring and reliability analysis. This paper investigates an uncertain dynamic load identification strategy with the combination of the Kalman filter (KF) algorithm and the random forest (RF) model. Enlightened by the concept of the KF, a one-step delayed identification of dynamic loads is completed in the state space domain. The previous loads can be calculated recursively utilizing the previous state estimation and the current displacement observation based on the least square method. In addition, the optimal state estimation will be achieved synchronously with the classical KF algorithm. In terms of inevitable uncertainties, the interval model is introduced to quantify uncertain system parameters and uncertain input loads. To determine the interval load process, the RF surrogate model is constructed to characterize their functional relationship, which is integrated by massive decision trees (DTs). Particularly, Latin hypercube sampling (LHS) and bootstrap sampling (BS) methods are employed to generate training samples for the RF model. The validity and feasibility of the developed methodology are eventually demonstrated by three numerical examples. The results indicate that the proposed approach has an excellent performance in both accuracy and efficiency.

White noise estimation for linear discrete fractional order system

&lt;abstract&gt; &lt;p&gt;The process white noise (PWN) and observation white noise (OWN) estimation problem for linear discrete fractional order systems (LDFOS) is addressed in this study. By using the Grünwald-Letnikov (G-L) operator as a definition of the discrete fractional calculus (DFC), LDFOS is transformed into a class of linear discrete time-delay systems. However, it is different from the general time-delay system, in which the time-delay part is the cumulative sum from time 0 to the previous time. Based on the orthogonal projection theorem, a suboptimal one-step predictor of LDFOS is designed. Due to the existence of cumulative sum time-delay in system, the Riccati equation has one more cumulative sum state error variance term, which is different from the classical Kalman filter (KF). Moreover, using innovation analysis technology, the filtering and fixed-lag smoothing estimators of PWN and OWN in the form of noise orthogonal projection gain matrices are derived. Finally, two simulation examples are given to verify the effectiveness of PWN and OWN estimators.&lt;/p&gt; &lt;/abstract&gt;

Application of Classical Kalman filtering technique in assimilation of multiple data types to NeQuick model

This study attempts to improve the estimation of ionospheric electron density profiles over Korea and adjacent areas by employing the classical Kalman filtering technique to assimilate Total Electron Content (TEC) data from various sources into the NeQuick model. Successive corrections method was applied to spread the effect of TEC data assimilation at a given location to others that lacked TEC observations. In order to reveal that the assimilation results emulate the complex ionospheric changes during geomagnetic storms, the selected study days included both quiet (Kp ≤ 3) and disturbed geomagnetic conditions in the year 2015. The results showed that assimilation of TEC data derived from ground-based Global Positioning System (GPS) receivers could improve the root mean squared error (RMSE) associated with the NeQuick model estimation of ionospheric parameters by ≥56%. The improvement of RMSE achieved by assimilating TEC data measured using ionosondes was ~50%. The assimilation of TEC observations made by the COSMIC radio occultation technique yielded results that depicted RMSE improvement of &gt;10%. The assimilation of TEC data measured by GPS receiver onboard Low Earth Orbiting satellites yielded results that revealed deterioration of RMSE. This outcome might be due to either the fact that the receivers are on moving platforms, and these dynamics might not have been accounted for during TEC computation or the limitation of the assimilation process. Validation of our assimilation results with global ionosphere TEC data maps as processed at the Center for Orbit Determination in Europe (CODE) revealed that both depicted similar TEC changes, showing response to a geomagnetic storm.

Contact force and torque sensing for serial manipulator based on an adaptive Kalman filter with variable time period

• Mode-switching moving average reduces current noise and improve torque confidence. • Time period adapts to achieve tradeoff between response time and precision. • Calibration routines take motor current noise and speed errors into account. • Adaptive Kalman filter with variable time period facilitates force/torque estimation. Contact force and torque sensing approaches enable manipulators to cooperate with humans and to interact appropriately with unexpected collisions. In this paper, a mode-switching moving average with variable time period is proposed to reduce the effects of measured motor current noise and thus provide improved confidence in joint output torque estimation. The time period of the filter adapts continuously to achieve optimal tradeoff between response time and precision of estimation in real-time. An adaptive Kalman filter that consists of the proposed moving average and the classical Kalman filter is proposed. Calibration routines for the adaptive Kalman filter take the measured motor current noise and errors in the speed data from the individual joints into account. The combination of the proposed adaptive Kalman filter with variable time period and its calibration method facilitates force and torque estimation without force/torque sensors. Contact force/torque sensing and response time assessments from the proposed approach were performed on the Universal Robot 5 manipulator with differing unexpected end effector loads. The combined force and torque sensing method led to a reduction of the estimation errors and response time in comparison with the pioneering method, and the effect is further improved as the payload rises. The proposed method can be applied to any robotic manipulators as long as the motor information (current, joint position, and joint velocities) are available and consequently the cost will be reduced dramatically from methods that require load cells.

Increasing the accuracy of the automatic control system of unmanned aircraft under conditions of non-stationary interference

An approach to solving the problem of improving the accuracy of an automatic control system for unmanned aerial vehicles (UAVs) in conditions of non-stationary interference is considered. Such interference is caused by random changes in the noise of the measuring sensors, which are installed in the UAV, when changing the yaw, roll and flight altitude of the UAV due to restructuring or transformation of the operating mode of the propulsion system of the UAV. Non-stationary noises do not allow obtaining the maximum effect of applying the theory of optimal filtration based on the classical Kalman filter. The authors proposed a method of stabilizing sensor noise by using a noise automatic gain control circuit installed in front of the Kalman filter. For its quantitative assessment, a team of authors developed a computer model and carried out dynamic modeling based on test signals, which showed the applicability of the classical Kalman filter in non-stationary conditions and a decrease in the filtering errors of the automatic control system.

Multiple-Model Estimation-based Prognostics for Rotating Machinery

Data-driven based prognostics approaches are currently attracting unprecedented attention. Considering flexibility, statistical model based prognostics, which present more transparency and usually incorporate statistical models with stochastic filtering, are studied in this paper. Traditionally, versions of Kalman filter and Particle filter are combined with a single statistical model. However, the service life of a bearing could undergo several health stages, such as a normal stage, a slight degradation stage, and a severe degradation stage. Thus, a single model cannot represent well the full degradation process of a bearing. As such, the concept of Multi-Model Estimation (MME) has been introduced in the field of CM of rotating machinery. Initially the Switching Kalman Filter (SKF) has been proposed combining three linear statistical models, but the Classic Kalman filter (CKF) might be insufficient for nonlinearity estimations. In this paper, we extend MME from CKF to nonlinear system estimation, for instance with extended Kalman filter, unscented Kalman filter and Particle filter, which can well overcome the drawbacks of CKF. Besides, twelve statistical models are studied and used for the automatic switch in the proposed multi-model methodology. The methodology is tested and evaluated on fifteen experimental datasets and it can be concluded that the extended MME outperforms the classic switching Kalman filter.

Centered error entropy Kalman filter with application to satellite attitude determination

Due to unavoidable factors, heavy-tailed noise appears in satellite attitude estimation. Traditional Kalman filter is prone to performance degradation and even filtering divergence when facing non-Gaussian noise. The existing robust algorithms have limited accuracy. To improve the attitude determination accuracy under non-Gaussian noise, we use the centered error entropy (CEE) criterion to derive a new filter named centered error entropy Kalman filter (CEEKF). CEEKF is formed by maximizing the CEE cost function. In the CEEKF algorithm, the prior state values are transmitted the same as the classical Kalman filter, and the posterior states are calculated by the fixed-point iteration method. The CEE EKF (CEE-EKF) algorithm is also derived to improve filtering accuracy in the case of the nonlinear system. We also give the convergence conditions of the iteration algorithm and the computational complexity analysis of CEEKF. The results of the two simulation examples validate the robustness of the algorithm we presented.

Adaptive algorithm for mobile robot movement control

The paper reviews the goal of controlling mobile robot movement along the planned path. It proposes an adaptive robot movement control algorithm, allowing to increase the precision of holding the robot within the prescribed path under the conditions of unstable navigation field. The adaptation of the algorithm to jamming environment is in redistributing weights of the weighted sum of estimates given by the ultrasound navigation system (USNS) and wheel pulse transducers (WPT). When precision of the USNS reduces affect of the WPT on the final estimate increases while affect of the USNS decreases. Such a redistribution is carried out when the system is operating due to changing elements of a measurement noise matrix in Kalman filter [1] with use of which a mobile robot heading angle is estimated. Corresponding elements of the measurement noise matrix increase along with increase of deviation of the mobile robot trajectory from a straight line. The extent of deviation calculated as the standard error of a regression is defined by means of the statistical analyses involving the last n measurements contained in buffer store of the USNS. The obtained results demonstrate significant benefit of the developed mobile robot heading estimation algorithm in comparison with the conventional filtering algorithm basing on using the classical Kalman filter with the measurement noise matrix whose elements are fixed.

A Generalized Identification of Joint Structural State and Unknown Inputs Using Data Fusion MKF-UI

The classical Kalman filter (KF) can estimate the structural state online in real time. However, the classical KF presupposes that external excitations are known. The existing methods of Kalman filter with unknown inputs (KF-UI) have limitations that require observing the acceleration response at the excitation point or assuming the unknown force. To surmount the above defects, an innovative modal Kalman filter with unknown inputs (MKF-UI) is proposed in this paper. Modal transformation and modal truncation are used to reduce the dimensionality of the structural state, and the accelerations at the excitation positions do not need to observe. Besides, the proposed MKF-UI does not require the assumption of unknown external excitation. Therefore, the proposed approach is suitable for the generalized identification of dynamic structural states and unknown loadings. The effectiveness and feasibility of the proposed identification approach are ascertained by some numerical simulation examples.

A simplified processing algorithm for multi-baseline RTK positioning in urban environments

The real-time kinematic (RTK) positioning with the multi-baseline solution (MBS) can provide high accuracy and availability for emerging applications. When the multi-systems and multi-frequencies observations for multiple stations are adopted, a heavy computational burden can arise to degrade the performance of RTK positioning. In this paper, a simplified multi-baseline solution (SMBS) algorithm based on the equivalence principle is proposed to improve the computational efficiency. Firstly, the classical equivalent observation equations (COE) are introduced and the variance–covariance (VC) matrix for SD observations is presented. Then, the simplified equivalent observation equations (SOE) can be obtained throughthe formuladerivationof transformation matrix. Secondly, the sequential Kalman filter (SKF) with multiple groups is used instead of the classical Kalman filter (CKF) to improve filtering efficiency. Finally, the performance of the proposed algorithm is comprehensively evaluated and analyzed by the static and kinematic experiments. The results demonstrate that the SMBS algorithm adopting SOE and SKF strategies can significantly accelerate the processing procedure of RTK positioning and reduce the processing time for a single epoch. The SOE strategy has an average efficiency improvement by approximately 74.7% for static, and 60.6% for the kinematic experiment, and the improvement of the SKF strategy is 49.6% and 58.0%. Moreover, the SMBS algorithm can provide a higher positioning accuracy and availability than the single-baseline solution (SBS) model. Especially for multiple reference stations, the state parameters of the rover station can be compressed to enhance the geometric strength of the positioning model, which can provide a smaller PDOP and ADOP values and improve the performance of ambiguity resolution (AR).

Hierarchical Dynamical Model for Multiple Cortical Neural Decoding.

Motor brain machine interfaces (BMIs) interpret neural activities from motor-related cortical areas in the brain into movement commands to control a prosthesis. As the subject adapts to control the neural prosthesis, the medial prefrontal cortex (mPFC), upstream of the primary motor cortex (M1), is heavily involved in reward-guided motor learning. Thus, considering mPFC and M1 functionality within a hierarchical structure could potentially improve the effectiveness of BMI decoding while subjects are learning. The commonly used Kalman decoding method with only one simple state model may not be able to represent the multiple brain states that evolve over time as well as along the neural pathway. In addition, the performance of Kalman decoders degenerates in heavy-tailed nongaussian noise, which is usually generated due to the nonlinear neural system or influences of movement-related noise in online neural recording. In this letter, we propose a hierarchical model to represent the brain states from multiple cortical areas that evolve along the neural pathway. We then introduce correntropy theory into the hierarchical structure to address the heavy-tailed noise existing in neural recordings. We test the proposed algorithm on in vivo recordings collected from the mPFC and M1 of two rats when the subjects were learning to perform a lever-pressing task. Compared with the classic Kalman filter, our results demonstrate better movement decoding performance due to the hierarchical structure that integrates the past failed trial information over multisite recording and the combination with correntropy criterion to deal with noisy heavy-tailed neural recordings.

Novel data fusion strategy for human gait analysis using multiple kinect sensors

Human gait analysis using microsoft Kinect sensor is an intriguing area of research. Fusing multiple sensors’ data for clinical gait analysis using calibrated instruments helps in producing more accurate results. A novel technique for spatial calibration is proposed and implemented in this paper. This paper emphasizes on a novel technique of using five Kinect V2 sensors (four clients and one server) and data fusion methods to create unified skeletons for subject-wise gait analysis. These eradicate the problems of inaccurate skeleton poses caused due to occlusions or tracking failures from a single Kinect which otherwise remains a problem in most vision-based sensing systems. Two methods are compared for estimating states of discrete time linear systems. The first one is classical Kalman filtering, which gives accurate results when state disturbances are assumed to be Gaussian white noises and measurement noises and the statistical properties are procurable. Secondly, a Set-Membership filter is used which relies upon the principle of prediction and correction as well. A novel Set-Membership filtering approach is proposed where the measurement noise is modelled by multivariate Gaussian probability density function bounded by −1 to +1. Based on our observations of the linear frameworks joined with interval investigation, the two phases of the estimator are done in a productive way. Both the fusion techniques are tested on overground and treadmill data and the results are validated and compared with ground truth obtained using Qualisys motion capture systems. The proposed approach is also compared quantitatively with state-of-the-art methods.

Structural-parametric synthesis of the tracking filter based on decomposition by the target functional with adaptation to trajectory disturbances

Traditional Kalman-type tracking filters are based on a kinematic motion model, which leads to the occurrence of dynamic errors, which significantly increase during target maneuvering. One of the solutions to this problem is to develop a model of motion dynamics with the ability to adapt its structure to external influences. It is shown that the use of a dynamic model of motion in the filter, which takes into account the inertia of the target and the forces acting on it, makes it possible to significantly increase the efficiency of the state assessment. Purpose is to development of an algorithm for assessing the position of a maneuvering object, effective in terms of accuracy criterion. The use of an adaptive motion model as part of the filter provides an increase in the estimation accuracy in comparison with the classical Kalman filter, which is confirmed by the performed numerical modeling.

A variational Bayesian-based robust adaptive filtering for precise point positioning using undifferenced and uncombined observations

In the application of precise point positioning (PPP), especially in the dynamic mode, the classical Kalman filter (KF) usually produces a large number of estimation errors or diverges when there are gross errors in the observation data or unexpected turbulences occur in target motion state or both of them. For such problem, a variational Bayesian (VB)-based robust adaptive Kalman filtering (VB-RAKF) is proposed in this paper. This filter introduces a classification robust equivalent weight function to resist observation gross error and the inverse Wishart prior to model inaccurate process noise covariance matrix (PNCM). To improve the instantaneous accuracy of state estimation, the VB approach is used to obtain better estimations of inaccurate PNCM. Several sets of observation data collected by IGS reference stations and vehicles are employed to check the robustness and positioning accuracy of the VB-RAKF model. The results show that the VB-RAKF algorithm is more robust than the KF, and can effectively resist the gross error in observation data and control state disturbance. In the IGS reference station tests, when compared to the KF, the static positioning accuracies of the VB-RAKF in the north, east and up directions are improved by 13%, 8% and 22%, respectively, and the simulated dynamic positioning accuracies of the VB-RAKF in the north, east and up directions are improved by 19%, 9% and 21%, respectively. The in-vehicle dynamic test verifies that the VB-RAKF outperforms the KF, and shows that the VB-RAKF has better performance than the KF when dealing with observation data which has obvious gross errors, and similar performance as the KF when gross errors are small.

Метод синтеза адаптивных алгоритмов оценки параметров динамических систем на основе принципа декомпозиции и методологии объединенного принципа максимума

A method of synthesis of a filter for estimating the state of dynamic systems of Kalman type with an adaptive model built on the basis of the principle of decomposition of the system using kinematic relations from the condition of constancy of motion invariants has been developed. The structure of the model is determined from the condition of the maximum function of the generalized power up to a nonlinear synthesizing function that determines the rate of dissipation and, accordingly, the degree of structural adaptation. The resulting model has an explicit relation with the gradient of the estimation error functional, which makes it possible to adapt to the intensity of regular and random influences and can be used to construct a filter for estimating the state of the Kalman structure. On the basis of the developed method, a discrete algorithm is obtained and its comparative analysis with the classical Kalman filter is carried out.