Augmented Extended Kalman Filter Research Articles

Using high sensitivity DC accelerometers for torque estimation on a wind turbine gearbox

In this paper we propose a novel approach to estimate the torque on a wind turbine gearbox using single-axis high sensitivity DC accelerometers. The torque estimator uses an Augmented Extended Kalman Filter (AEKF), combining physics-based models with measurement data. The accelerometer sensor model is derived by analysing the measurements, leading to the identification of two torque-driven acceleration contributions: acceleration due to force excitation and acceleration due to a change in measurement direction. For the latter, two different sources, with distinct frequency content, are capable of changing the measurement direction. The first source corresponds to rigid body motion (f < 0.15Hz), the second one to ring gear deformation (0.15 < f < 4Hz). Using this information, the measured signal can be filtered to target specific acceleration contributions. The first torque estimation approach targets frequencies from 0.15 to 4Hz and produces a Normalised Mean Absolute Error (NMAE) of 2.12%. The second approach targets frequencies up to 0.15Hz, and will estimate the rigid body deflection angle. The torque is calculated through a linear identified relation between the rigid body angle and applied input torque. This approach yields an NMAE of 2.59%. This leads to the conclusion that high sensitivity DC accelerometers are a valid option for estimating torque on a wind turbine gearbox. These torque monitoring approaches will pave the way for advanced condition monitoring strategies and the calculation of remaining useful lifetime of the gearbox.

Exploiting Cyclic Angle-Dependency in a Kalman Filter-Based Torque Estimation on a Mechatronic Drivetrain

Torsional vibrations play a critical role in the design and operation of a mechanical or mechatronic drivetrain due to their impact on lifetime, performance, and cost. A magnetic spring allows one to reduce these vibrations and improve the actuator performance yet introduces additional challenges on the identification. As a direct torque measurement is generally not favourable because of its intrusive nature, this paper proposes a nonintrusive approach to identify torsional load profiles. The approach combines a physics-based lumped parameter model of the torsional dynamics of the drivetrain with measurements coming from a motor encoder and two MEMS accelerometers in a combined state/input estimation, using an augmented extended Kalman filter (A-EKF). In order to allow a generic magnetic spring torque estimation, a random walk input model is used, where additionally the angle-dependent behaviour is exploited by constructing an angle-dependent estimate and variance map. Experimental validation leads to a significant reduction in bias in the load torque estimation for this approach, compared to conventional estimators. Moreover, this newly proposed approach significantly reduces the variance on the estimated states by exploiting the angle dependency. The proposed approach provides knowledge of the torsional vibrations in a nonintrusive way, without the need for an extensive magnetic spring torque identification. Further, the approach is applicable on any drivetrain with angle-dependent input torques.

Sensor selection for cost-effective virtual torque measurements on a wind turbine gearbox

Measuring the operational torques during the lifetime of a wind turbine gearbox offers a valuable source of information for design, monitoring, predictive maintenance and control, and can help reduce operational expenses and downtime. In this paper we investigate indirect sensing as a valuable alternative to direct sensing, which is costly and intrusive. The resulting virtual torque sensor makes use of an Augmented Extended Kalman filter (AEKF), a physics based model, and measurements. The measurements considered are encoders on the high- and low speed shaft together with strain sensors, which are easy to install on the gearbox housing. We first discuss the theoretical background of the AEKF and methods for tuning the AEKF parameters. We then give an overview on the models used. Next, we focus on a sensor selection algorithm that can select the most performant (e.g. high signal-to-noise ratio) sensors. Finally, we validate the virtual torque sensor numerically. We show that with encoder measurements on the high speed shaft and 7 strain gauges placed on the housing we can achieve a Normalized Root Mean Squared Error (NRMSE) of 1.27% and 0.02% for the low- and high speed shaft torque, respectively. Without any strain gauges (i.e., only with the encoders), the estimation accuracy degrades to an NRMSE of 1.37% and 0.12%. A further analysis of the estimation results with different amounts of strain gauges shows that the amount of strain gauges can be reduced without a significant loss in estimation accuracy. These results show the potential of virtual torque sensing as a mean to obtain operational torque information for the design, monitoring and control of wind turbine gearboxes.

Expectation‐maximization‐based infrared target tracking with time‐varying extinction coefficient identification

SummaryExtinction coefficient (EC), as the key parameter of target intensity model, is assumed constant in classical infrared target tracking (IRTT) methods. However, it is a time‐varying and state‐coupled parameter related to complex atmosphere environment. To this end, this article proposes the problem of IRTT with time‐varying EC identification. Different from the constant EC case whose solution is the measurement augmentation, the time‐varying EC case brings out the new challenge: deep coupling between state estimation and parameter identification. In the expectation‐maximization framework, this article derives the joint identification and estimation optimization scheme, where the Taylor expansion variance error of intensity model is also identified to adaptively compensate the nonlinear approximation. Simulation examples show that the proposed scheme has better estimation accuracy than the existing augmented extended Kalman filter.

Augmented extended Kalman filter with cooperative Bayesian filtering and multi‐models fusion for precise vehicle localisations

Self-localisation is vital for autonomous vehicles. In this study, the authors present an augmented extended Kalman filter (AEKF) framework for intelligent vehicle localisation applications. Compared to the previous approach, the proposed AEKF is enhanced through a model fusion, which incorporates a constant velocity model, constant acceleration model, constant turn rate and velocity, and constant turn rate and acceleration model by using the Takagi–Sugeno fuzzy inference technique, where the typical prediction procedure in the extended Kalman filter is modified by a fusion of those various motion models for the state estimation. Furthermore, they proposed a flexible cooperative Bayesian filter to incorporate the data from nearby-vehicles’ position and lateral distance from the host vehicle to the lane lines, to improve the raw global positioning system (GPS) performance under multi-sensor observation environments. They conduct simulation experiments under vividly, near-realistic scenarios with random traffic-flows to show the superiorities of the proposed framework when compared with the consumer-grade GPS implementation. The results show that the obtained positioning enhancement can significantly reduce the positioning error from the original larger than 5 m to the sub-meter level under various scenarios.

Recursive principal component analysis for model order reduction with application in nonlinear Bayesian filtering

Proper orthogonal decomposition (POD) is a useful technique for feature extraction, model order reduction and data compression and has been widely used in different science and engineering disciplines. Numerous papers have been published on the application of offline POD, i.e., batch POD (BPOD) in civil and mechanical engineering encompassing Karhunen–Loève decomposition (KLD), principal component analysis (PCA), and singular value decomposition (SVD). Nevertheless, online POD which is more suited for online feature extraction and monitoring has been scarcely addressed when dealing with civil and mechanical systems, particularly in structural dynamics. In this paper, a number of recursive POD (RPOD) methods in form of recursive PCA (RPCA) are overviewed with their application to structural dynamics. RPCA with numerical eigenvalue decomposition (EVD), incremental principal component analysis (IPCA), matrix perturbation method, and Kalman filter RPCA (KFRPCA) are presented; their performance is probed in terms of initialization, structural parameter modification, noisy observation, and alteration of loading statistics. The novel KFRPCA algorithm developed in this paper is reformulated to resolve the unobservability issue of higher modes which was present in its previous version in the published literature. Online stochastic output-only system identification is presented by synergizing RPCA with nonlinear Bayesian filter. Augmented extended Kalman filter (AEKF) is employed to perform unknown-input dual estimation.

Polynomial Augmented Extended Kalman Filter to Estimate the State of Charge of Lithium-Ion Batteries

In battery-electric vehicles, an accurate knowledge of the current state of charge (SOC) of the battery is crucial for safe and efficient operation. Offset-free SOC estimation for Lithium-ion batteries (LIB) during operation still is a challenging problem, due to the uncertain output nonlinearity present in battery models. In battery management systems employed in such vehicles, the age- and cycle-dependent relationship between the open circuit voltage (OCV) and the state of charge of each cell (SOC) can be kept track of using lookup tables. Between the scattered data points, linear interpolation is often used. Another common approach is to switch between models, each with a different piecewise polynomial fit of low order. In this contribution, a single polynomial fit of a higher order with state-dependent coefficients for the whole SOC-OCV range is proposed, so as to avoid switching and facilitate usage of Taylor-based linearization algorithms like in the extended Kalman filter. While the polynomial order is high, the number of parameters is only three; the parameters are estimated and updated by the Kalman filter itself. This augmentation of the observer's state vector with said polynomial parameters is inspired by the idea of an increased “stochastization,” introducing redundancy that aids to achieve a more accurate SOC estimation. In this sense, the algorithm can be described as a model-adaptive extended Kalman estimator. Two variants of a polynomial EKF are investigated: polynomial EKF with output and with state nonlinearity models. Their performances are compared in real experiments using a dedicated test bench based on a Samsung ICR18650-26F cell, demonstrating the effectiveness of the algorithms. The variant that includes the terminal voltage in the state vector, i.e. the state nonlinearity variant of the EKF, shows better result, also compared to the existing literature.

Relative sensor registration with two‐step method for state estimation

State estimation suffers from some new challenging problems with a multi-platform multi-sensor observation system. An important problem for multisensor integration is that the data from the local sensors needs to be transformed into a common reference frame free of systematic bias or registration. In this study, the relative sensor registration problem is discussed. It aligns measurement from the global sensor with the local sensor under the assumptions that the global sensor is bias free and all biases reside with the local sensor. The traditional methods failed in the condition when attitude bias becomes large because the error caused by linearisation of rotation matrix increases with growing attitude bias. Motivated by this, a two-step method is established. By estimating the measurement bias through augmented extended Kalman filter in local sensor coordinate independent of attitude and location bias, and by introducing the unit quaternion method compute the attitude and location bias, the proposed method can avoid the problem the traditional methods faced. Simulation examples are provided to verify the proposed method by comparing with the existing linear least square algorithm.

Sensor-fault tolerant attitude determination using two-stage estimator

Satellite attitude determination accuracy significantly drops when sensor-fault occurs. Hence, a proper mitigation strategy to detect sensor-fault and accurately estimate corresponding fault magnitudes is mandatory for robust and accurate attitude determination. In this paper, a novel sensor-fault tolerant precise attitude estimator is proposed consisting of two stages. In the first stage, sensor-fault is detected, and the associated sensor parameter change is roughly estimated using an interacting multiple-model (IMM) approach. Subsequently, the second stage is triggered. The sensor parameter change is precisely estimated with a new sensor-parameter-augmented filter. This is defined as a selectively augmented extended Kalman filter (SAEKF) in this paper. The conventional augmented extended Kalman filter (AEKF) is computationally more expensive than the proposed SAEKF. The SAEKF augments only the sensor parameters affected by sensor-faults, not the full sensor parameters, into the state vector. This leads to a significant computational time-saving. A transition method from the first stage to the second stage is also investigated. Numerical simulation results demonstrate that the proposed two-stage approach has smaller attitude determination errors than the existing algorithms, ranged from 21.7% to 88.8%, in cases with gyro scale factor error or misalignment.

Multibody model based estimation of multiple loads and strain field on a vehicle suspension system

Abstract This work proposes an augmented extended Kalman filter based state-input estimator for mechanical systems defined by implicit equations of motion which is then applied to estimate the six wheel center loads and the strain field on a vehicle suspension test rig. Implicit equations of motion typically arise in the definition of flexible multibody models and also in their time resolution, because implicit time-discretization schemes are normally employed to obtain a stable solution. The presented methodology can be applied to such case and analytical expressions are derived for the necessary linearizations, providing the means for a computationally efficient estimation procedure. The six wheel center loads and the strain field on a vehicle suspension system are valuable quantities during the vehicle design phase (e.g. for durability analysis), hence they are often directly measured during elaborate full vehicle testing campaigns. This work demonstrates that a flexible multibody model representation allows to accurately reconstruct the time domain signals of the six loads and of the full strain field, starting from a minimal set of six measured strains, hence providing an appealing alternative to direct measurement methods. The experimental validation on the suspension test rig shows that all estimated quantities can be accurately reconstructed, given that the system simulation model incorporates an adequate level of accuracy.

Vehicle dynamics estimation via augmented Extended Kalman Filtering

The response of active safety systems of modern cars strongly depends on the estimation accuracy in the key motion states of the vehicle. One common limitation of current systems is the lack of adaptability in the parameters of the vehicle model that are usually treated as time-invariant, although they are not exactly known or are subject to temporal changes. As a direct consequence, time invariant-parameter control systems may achieve sub-optimal performance and/or deteriorate according to the driving conditions.This paper presents a non-linear model-based observer for combined estimation of motion states and tyre cornering stiffness. It is based on common onboard sensors, that is a lateral acceleration and yaw rate sensor, and it works during normal vehicle manoeuvering. The identification framework relies on an augmented Extended Kalman filter to deal with model parameter variability and noisy measurement input. Results are described to evaluate the performance and sensitivity of the proposed approach, showing an improvement in the estimation accuracy that can reach an order of magnitude compared to standard approaches.

AEKF-SLAM: A New Algorithm for Robotic Underwater Navigation.

In this work, we focus on key topics related to underwater Simultaneous Localization and Mapping (SLAM) applications. Moreover, a detailed review of major studies in the literature and our proposed solutions for addressing the problem are presented. The main goal of this paper is the enhancement of the accuracy and robustness of the SLAM-based navigation problem for underwater robotics with low computational costs. Therefore, we present a new method called AEKF-SLAM that employs an Augmented Extended Kalman Filter (AEKF)-based SLAM algorithm. The AEKF-based SLAM approach stores the robot poses and map landmarks in a single state vector, while estimating the state parameters via a recursive and iterative estimation-update process. Hereby, the prediction and update state (which exist as well in the conventional EKF) are complemented by a newly proposed augmentation stage. Applied to underwater robot navigation, the AEKF-SLAM has been compared with the classic and popular FastSLAM 2.0 algorithm. Concerning the dense loop mapping and line mapping experiments, it shows much better performances in map management with respect to landmark addition and removal, which avoid the long-term accumulation of errors and clutters in the created map. Additionally, the underwater robot achieves more precise and efficient self-localization and a mapping of the surrounding landmarks with much lower processing times. Altogether, the presented AEKF-SLAM method achieves reliably map revisiting, and consistent map upgrading on loop closure.

Extended input estimation method for tracking non‐linear manoeuvring targets with multiplicative noises

In this study, a new input estimation method is proposed for targets with non-linear dynamics and unknown inputs (manoeuvres), where measurement model is corrupted with both additive and multiplicative noises. First, the authors proposed an innovative model by adding unknown inputs as a new state to the original state vector and constructed an augmented state vector. Therefore, manoeuvring model turns into a non-manoeuvring model. The proposed model does not need any manoeuvre detection stage procedure but it causes a correlation between the process and measurement noises. Subsequently, the authors modify the augmented model to deal with the cross-correlation problem. Finally an augmented extended Kalman filter scheme is proposed, which unlike the most existing literature, it can estimate unknown inputs as quickly as possible and copes with the multiplicative noises and the cross-correlation problem. The efficiency of the proposed method is demonstrated in computer simulations for a manoeuvring target with non-linear dynamics through Monte-Carlo simulation.

An Improved Otsu Threshold Segmentation Method for Underwater Simultaneous Localization and Mapping-Based Navigation.

The main focus of this paper is on extracting features with SOund Navigation And Ranging (SONAR) sensing for further underwater landmark-based Simultaneous Localization and Mapping (SLAM). According to the characteristics of sonar images, in this paper, an improved Otsu threshold segmentation method (TSM) has been developed for feature detection. In combination with a contour detection algorithm, the foreground objects, although presenting different feature shapes, are separated much faster and more precisely than by other segmentation methods. Tests have been made with side-scan sonar (SSS) and forward-looking sonar (FLS) images in comparison with other four TSMs, namely the traditional Otsu method, the local TSM, the iterative TSM and the maximum entropy TSM. For all the sonar images presented in this work, the computational time of the improved Otsu TSM is much lower than that of the maximum entropy TSM, which achieves the highest segmentation precision among the four above mentioned TSMs. As a result of the segmentations, the centroids of the main extracted regions have been computed to represent point landmarks which can be used for navigation, e.g., with the help of an Augmented Extended Kalman Filter (AEKF)-based SLAM algorithm. The AEKF-SLAM approach is a recursive and iterative estimation-update process, which besides a prediction and an update stage (as in classical Extended Kalman Filter (EKF)), includes an augmentation stage. During navigation, the robot localizes the centroids of different segments of features in sonar images, which are detected by our improved Otsu TSM, as point landmarks. Using them with the AEKF achieves more accurate and robust estimations of the robot pose and the landmark positions, than with those detected by the maximum entropy TSM. Together with the landmarks identified by the proposed segmentation algorithm, the AEKF-SLAM has achieved reliable detection of cycles in the map and consistent map update on loop closure, which is shown in simulated experiments.

Augmented consensus filter for simultaneous localization and tracking with limited sense range

In this paper, we investigate how to exploit distributed average consensus fusion for conducting simultaneous localization and tracking (SLAT) by using wireless sensor networks. To this end, we commence by establishing a limited sense range (LSR) nonlinear system that characterizes the coupling of target state and sensor localization with respect to each sensor. We then employ an augmented extended Kalman filter to estimate the sensor and target states of our system. Furthermore, we adopt a consensus filtering scheme which fuses the information from neighboring sensors. We thus obtain a two-stage distributed filtering framework that not only obtains updated sensor locations trough augment filtering but also provides an accurate target state estimate in consensus filtering. Additionally, our framework is computationally efficient because it only requires neighboring sensor communications. The simulation results reveal that the proposed filtering framework is much more robust than traditional information fusion methods in limited ranging conditions.

An Enhanced Error Model for EKF-Based Tightly-Coupled Integration of GPS and Land Vehicle's Motion Sensors.

Reduced inertial sensor systems (RISS) have been introduced by many researchers as a low-cost, low-complexity sensor assembly that can be integrated with GPS to provide a robust integrated navigation system for land vehicles. In earlier works, the developed error models were simplified based on the assumption that the vehicle is mostly moving on a flat horizontal plane. Another limitation is the simplified estimation of the horizontal tilt angles, which is based on simple averaging of the accelerometers’ measurements without modelling their errors or tilt angle errors. In this paper, a new error model is developed for RISS that accounts for the effect of tilt angle errors and the accelerometer’s errors. Additionally, it also includes important terms in the system dynamic error model, which were ignored during the linearization process in earlier works. An augmented extended Kalman filter (EKF) is designed to incorporate tilt angle errors and transversal accelerometer errors. The new error model and the augmented EKF design are developed in a tightly-coupled RISS/GPS integrated navigation system. The proposed system was tested on real trajectories’ data under degraded GPS environments, and the results were compared to earlier works on RISS/GPS systems. The findings demonstrated that the proposed enhanced system introduced significant improvements in navigational performance.

Augmented Filtering Based on Information Weighted Consensus Fusion for Simultaneous Localization and Tracking via Wireless Sensor Networks

This paper develops a novel augmented filtering framework based on information weighted consensus fusion, to achieve the simultaneous localization and tracking (SLAT) via wireless sensor networks (WSNs). By integrating augmented transition and observation models, we formulate a dynamical system that encodes both the target moving manners and coarse sensor locations in an augmented state. We then conduct augmented filtering based on augmented extended Kalman filters to estimate the augmented state. We further refine our target estimate according to information weighted consensus filtering which fuses the target information obtained from neighboring sensors. The fused information is fed back as the target estimate to the augmented filter. Our framework is computationally efficient because it only requires neighboring sensor communications. Experiments on SLAT problem validate the effectiveness of the proposed algorithm in terms of tracking accuracy and localization precision in limited ranging conditions.

Global registration of subway tunnel point clouds using an augmented extended Kalman filter and central-axis constraint.

Because tunnels generally have tubular shapes, the distribution of tie points between adjacent scans is usually limited to a narrow region, which makes the problem of registration error accumulation inevitable. In this paper, a global registration method is proposed based on an augmented extended Kalman filter and a central-axis constraint. The point cloud registration is regarded as a stochastic system, and the global registration is considered to be a process that recursively estimates the rigid transformation parameters between each pair of adjacent scans. Therefore, the augmented extended Kalman filter (AEKF) is used to accurately estimate the rigid transformation parameters by eliminating the error accumulation caused by the pair-wise registration. Moreover, because the scanning range of a terrestrial laser scanner can reach hundreds of meters, a single scan can cover a tunnel segment with a length of more than one hundred meters, which means that the central axis extracted from the scan can be employed to control the registration of multiple scans. Therefore, the central axis of the subway tunnel is first determined through the 2D projection of the tunnel point cloud and curve fitting using the RANSAC (RANdom SAmple Consensus) algorithm. Because the extraction of the central axis by quadratic curve fitting may suffer from noise in the tunnel points and from variations in the tunnel, we present a global extraction algorithm that is based on segment-wise quadratic curve fitting. We then derive the central-axis constraint as an additional observation model of AEKF to optimize the registration parameters between each pair of adjacent scans. The proposed approach is tested on terrestrial point clouds that were acquired in a subway tunnel. The results show that the proposed algorithm is capable of improving the accuracy of aligning multiple scans by 48%.

Distributed Widely Linear Kalman Filtering for Frequency Estimation in Power Networks

Motivated by the growing need for robust and accurate frequency estimators at the low- and medium-voltage distribution levels and the emergence of ubiquitous sensors networks for the smart grid, we introduce a distributed Kalman filtering scheme for frequency estimation. This is achieved by using widely linear state space models, which are capable of estimating the frequency under both balanced and unbalanced operating conditions. The proposed distributed augmented extended Kalman filter (D-ACEKF) exploits multiple measurements without imposing any constraints on the operating conditions at different parts of the network, while also accounting for the correlated and noncircular natures of real-world nodal disturbances. Case studies over a range of power system conditions illustrate the theoretical and practical advantages of the proposed methodology.

Consider unobservable uncertain parameters using radio beacon navigation during Mars entry

This paper presents a navigation algorithm based on the extended consider Kalman filter (ECKF) to mitigate the adverse effects of unobservable uncertain parameters using radio measurements during Mars entry phase. Consistency of the Mars entry navigation using a radio beacon navigation scheme is also demonstrated. A nonlinear consider approach is utilized by incorporating the covariances of the unobservable uncertain atmospheric density and the ballistic coefficient into the state estimate covariance; parameters are specifically not estimated directly to improve the navigation accuracy and robustness during the Mars entry phase. Perturbation is introduced to analyze the navigation state errors from each unobservable uncertain parameter. Numerical simulations show that the presented ECKF-based navigation algorithm is more accurate and robust than the augmented extended Kalman filter for unobservable uncertain parameters.