Showing papers on "Invariant extended Kalman filter published in 2013"

High-precision, consistent EKF-based visual-inertial odometry

Abstract: In this paper, we focus on the problem of motion tracking in unknown environments using visual and inertial sensors. We term this estimation task visual-inertial odometry (VIO), in analogy to the well-known visual-odometry problem. We present a detailed study of extended Kalman filter (EKF)-based VIO algorithms, by comparing both their theoretical properties and empirical performance. We show that an EKF formulation where the state vector comprises a sliding window of poses (the multi-state-constraint Kalman filter (MSCKF)) attains better accuracy, consistency, and computational efficiency than the simultaneous localization and mapping (SLAM) formulation of the EKF, in which the state vector contains the current pose and the features seen by the camera. Moreover, we prove that both types of EKF approaches are inconsistent, due to the way in which Jacobians are computed. Specifically, we show that the observability properties of the EKF's linearized system models do not match those of the underlying system, which causes the filters to underestimate the uncertainty in the state estimates. Based on our analysis, we propose a novel, real-time EKF-based VIO algorithm, which achieves consistent estimation by (i) ensuring the correct observability properties of its linearized system model, and (ii) performing online estimation of the camera-to-inertial measurement unit (IMU) calibration parameters. This algorithm, which we term MSCKF 2.0, is shown to achieve accuracy and consistency higher than even an iterative, sliding-window fixed-lag smoother, in both Monte Carlo simulations and real-world testing.

Evaluation on State of Charge Estimation of Batteries With Adaptive Extended Kalman Filter by Experiment Approach

Abstract: An accurate State-of-Charge (SoC) estimation plays a significant role in battery systems used in electric vehicles due to the arduous operation environments and the requirement of ensuring safe and reliable operations of batteries. Among the conventional methods to estimate SoC, the Coulomb counting method is widely used, but its accuracy is limited due to the accumulated error. Another commonly used method is model-based online iterative estimation with the Kalman filters, which improves the estimation accuracy in some extent. To improve the performance of Kalman filters in SoC estimation, the adaptive extended Kalman filter (AEKF), which employs the covariance matching approach, is applied in this paper. First, we built an implementation flowchart of the AEKF for a general system. Second, we built an online open-circuit voltage (OCV) estimation approach with the AEKF algorithm so that we can then get the SoC estimate by looking up the OCV-SoC table. Third, we proposed a robust online model-based SoC estimation approach with the AEKF algorithm. Finally, an evaluation on the SoC estimation approaches is performed by the experiment approach from the aspects of SoC estimation accuracy and robustness. The results indicate that the proposed online SoC estimation with the AEKF algorithm performs optimally, and for different error initial values, the maximum SoC estimation error is less than 2% with close-loop state estimation characteristics.

A Student's t filter for heavy tailed process and measurement noise

Abstract: We consider the filtering problem in linear state space models with heavy tailed process and measurement noise. Our work is based on Student's t distribution, for which we give a number of useful results. The derived filtering algorithm is a generalization of the ubiquitous Kalman filter, and reduces to it as special case. Both Kalman filter and the new algorithm are compared on a challenging tracking example where a maneuvering target is observed in clutter.

Feedback Particle Filter

Abstract: The feedback particle filter introduced in this paper is a new approach to approximate nonlinear filtering, motivated by techniques from mean-field game theory. The filter is defined by an ensemble of controlled stochastic systems (the particles). Each particle evolves under feedback control based on its own state, and features of the empirical distribution of the ensemble. The feedback control law is obtained as the solution to an optimal control problem, in which the optimization criterion is the Kullback-Leibler divergence between the actual posterior, and the common posterior of any particle. The following conclusions are obtained for diffusions with continuous observations: 1) The optimal control solution is exact: The two posteriors match exactly, provided they are initialized with identical priors. 2) The optimal filter admits an innovation error-based gain feedback structure. 3) The optimal feedback gain is obtained via a solution of an Euler-Lagrange boundary value problem; the feedback gain equals the Kalman gain in the linear Gaussian case. Numerical algorithms are introduced and implemented in two general examples, and a neuroscience application involving coupled oscillators. In some cases it is found that the filter exhibits significantly lower variance when compared to the bootstrap particle filter.

Estimation of the Dynamic States of Synchronous Machines Using an Extended Particle Filter

Abstract: In this paper, an extended particle filter (PF) is proposed to estimate the dynamic states of a synchronous machine using phasor measurement unit (PMU) data. A PF propagates the mean and covariance of states via Monte Carlo simulation, is easy to implement, and can be directly applied to a nonlinear system with non-Gaussian noise. The proposed extended PF improves robustness of the basic PF through iterative sampling and inflation of particle dispersion. Using Monte Carlo simulations with practical noise and model uncertainty considerations, the extended PF's performance is evaluated and compared with the basic PF, an extended Kalman filter (EKF) and an unscented Kalman filter (UKF). The extended PF results showed high accuracy and robustness against measurement and model noise.

State Space Models and MIDAS Regressions

Abstract: We examine the relationship between Mi(xed) Da(ta) S(ampling) (MIDAS) regressions and the Kalman filter when forecasting with mixed frequency data. In general, state space models involve a system of equations, whereas MIDAS regressions involve a single equation. As a consequence, MIDAS regressions might be less efficient, but could also be less prone to parameter estimation error and/or specification errors. We examine how MIDAS regressions and Kalman filters match up under ideal circumstances, that is in population, and in cases where all the stochastic processes—low and high frequency—are correctly specified. We characterize cases where the MIDAS regression exactly replicates the steady state Kalman filter weights. We compare MIDAS and Kalman filter forecasts in population where the state space model is misspecified. We also compare MIDAS and Kalman filter forecasts in small samples. The paper concludes with an empirical application. Overall we find that the MIDAS and Kalman filter methods give similar ...

Transformed Unscented Kalman Filter

Abstract: This technical note concerns the deterministic sampling points construction strategy for unscented Kalman filter (UKF) and cubature Kalman filter (CKF). From the numerical-integration viewpoint, a new deterministic sampling points set is derived by orthogonal transformation on the cubature points. By embedding these points into the UKF framework, a modified nonlinear filter named transformed unscented Kalman filter (TUKF) is derived. The TUKF can address the nonlocal sampling problem inherent in CKF while maintaining the virtue of numerical stability for high dimensional problems. Moreover, the methodology proposed in this technical note can be used to construct nonlinear filters with improved accuracy for certain problems. The performance of the proposed algorithm is demonstrated through a nonlinear high dimensional problem.

Hybrid Kalman Filters for Very Short-Term Load Forecasting and Prediction Interval Estimation

Abstract: Very short-term load forecasting predicts the loads in electric power system one hour into the future in 5-min steps in a moving window manner. To quantify forecasting accuracy in real-time, the prediction interval estimates should also be produced online. Effective predictions with good prediction intervals are important for resource dispatch and area generation control, and help power market participants make prudent decisions. We previously presented a two level wavelet neural network method based on back propagation without estimating prediction intervals. This paper extends the previous work by using hybrid Kalman filters to produce forecasting with prediction interval estimates online. Based on data analysis, a neural network trained by an extended Kalman filter is used for the low-low frequency component to capture the near-linear relationship between the input load component and the output measurement, while neural networks trained by unscented Kalman filters are used for low-high and high frequency components to capture their nonlinear relationships. The overall variance estimate is then derived and evaluated for prediction interval estimation. Testing results demonstrate the effectiveness of hybrid Kalman filters for capturing different features of load components, and the accuracy of the overall variance estimate derived based on a data set from ISO New England.

Optimal Linear Filters for Discrete-Time Systems With Randomly Delayed and Lost Measurements With/Without Time Stamps

Abstract: A novel model is developed to describe possible random delays and losses of measurements transmitted from a sensor to a filter by a group of Bernoulli distributed random variables. Based on the new developed model, an optimal linear filter dependent on the probabilities is presented in the linear minimum variance sense by the innovation analysis approach when packets are not time-stamped. The solution to the optimal linear filter is given in terms of a Riccati difference equation and a Lyapunov difference equation. A sufficient condition for the existence of the steady-state filter is given. At last, the optimal filter is given by Kalman filter when packets are time-stamped.

Kalman Filter-Based Distributed Predictive Control of Large-Scale Multi-Rate Systems: Application to Power Networks

Abstract: In this paper, a novel distributed Kalman filter (KF) algorithm along with a distributed model predictive control (MPC) scheme for large-scale multi-rate systems is proposed. The decomposed multi-rate system consists of smaller subsystems with linear dynamics that are coupled via states. These subsystems are multi-rate systems in the sense that either output measurements or input updates are not available at certain sampling times. Such systems can arise, e.g., when the number of sensors is smaller than the number of variables to be controlled, or when measurements of outputs cannot be completed simultaneously because of practical limitations. The multi-rate nature gives rise to lack of information, which will cause uncertainty in the system's performance. To circumvent this problem, we propose a distributed KF-based MPC scheme, in which multiple control and estimation agents each determine actions for their own parts of the system. Via communication, the agents can in a cooperative way take one another's actions into account. The main task of the proposed distributed KF is to compensate for the information loss due to the multi-rate nature of the systems by providing optimal estimation of the missing information. A demanding two-area power network example is used to demonstrate the effectiveness of the proposed method.

Square Root Cubature Information Filter

Abstract: Nonlinear state estimation plays a major role in many real-life applications. Recently, some sigma-point filters, such as the unscented Kalman filter, the particle filter, or the cubature Kalman filter have been proposed as promising substitutes for the conventional extended Kalman filter. For multisensor fusion, the information form of the Kalman filter is preferred over standard covariance filters due to its simpler measurement update stage. This paper presents a new state estimation algorithm called the square root cubature information filter (SRCIF) for nonlinear systems. The cubature information filter is first derived from an extended information filter and a recently developed cubature Kalman filter. For numerical accuracy, its square root version is then developed. Unlike the extended Kalman or extended information filters, the proposed filter does not require the evaluation of Jacobians during state estimation. The proposed approach is further extended for use in multisensor state estimation. The efficacy of the SRCIF is demonstrated by a simulation example of a permanent magnet synchronous motor.

A Gaussian-Sum Based Cubature Kalman Filter for Bearings-Only Tracking

Abstract: Herein is presented an efficient nonlinear filtering algorithm called the Gaussian-sum cubature Kalman filter (GSCKF) for the bearings-only tracking problem. It is developed based on the recently proposed cubature Kalman filter and is built within a Gaussian-sum framework. The new algorithm consists of a splitting and merging procedure when a high degree of nonlinearity is detected. Simulation results show that the proposed algorithm demonstrates comparable performance to the particle filter (PF) with significantly reduced computational cost.

Kalman Filter-Based Integrity Monitoring Against Sensor Faults

Abstract: This paper introduces a new Kalman filter-based method for detecting sensor faults in linear dynamic systems. In contrast with existing sequential fault-detection algorithms, the proposed method enables direct evaluation of the integrity risk, which is the probability that an undetected fault causes state estimate errors to exceed predefined bounds of acceptability. The new method is also computationally efficient and straightforward to implement. The algorithm’s detection test statistic is established in three steps. First, the weighted norms of current and past-time Kalman filter residuals are defined as generalized noncentrally chi-square distributed random variables. Second, these residuals are proven to be stochastically independent from the state estimate error. Third, current-time and past-time residuals are shown to be mutually independent, so that the Kalman filter-based test statistic can be recursively updated in real time by simply adding the current-time residual contribution to a previously ...

Stochastic Integration Filter

Abstract: The technical note deals with state estimation of nonlinear stochastic dynamic systems. Traditional filters providing local estimates of the state, such as the extended Kalman filter, unscented Kalman filter, or the cubature Kalman filter, are based on computationally efficient but approximate integral evaluations. On the other hand, the Monte Carlo based Kalman filter takes an advantage of asymptotically exact integral evaluations but at the expense of substantial computational demands. The aim of the technical note is to propose a new local filter that utilises stochastic integration methods providing the asymptotically exact integral evaluation with computational complexity similar to the traditional filters. The technical note will demonstrate that the unscented and cubature Kalman filters are special cases of the proposed stochastic integration filter. The proposed filter is illustrated by a numerical example.

Study of the General Kalman Filter for Echo Cancellation

Abstract: The Kalman filter is a very interesting signal processing tool, which is widely used in many practical applications. In this paper, we study the Kalman filter in the context of echo cancellation. The contribution of this work is threefold. First, we derive a different form of the Kalman filter by considering, at each iteration, a block of time samples instead of one time sample as it is the case in the conventional approach. Second, we show how this general Kalman filter (GKF) is connected with some of the most popular adaptive filters for echo cancellation, i.e., the normalized least-mean-square (NLMS) algorithm, the affine projection algorithm (APA) and its proportionate version (PAPA). Third, a simplified Kalman filter is developed in order to reduce the computational load of the GKF; this algorithm behaves like a variable step-size adaptive filter. Simulation results indicate the good performance of the proposed algorithms, which can be attractive choices for echo cancellation.

How To NOT Make the Extended Kalman Filter Fail

Abstract: In an effort to assess the performance of newer estimation algorithms, many prior publications have presented comparative studies where the Extended Kalman Filter (EKF) failed. This is because the ...

Novel Model-Based Estimators for the Purposes of Fault Detection and Diagnosis

Abstract: The interacting multiple model (IMM) strategy is particularly useful for systems that behave according to a number of different operating modes. In this strategy, each operating mode is described by a model and has its own filter. The filters are run in parallel, and an overall operating mode probability is calculated that provides an indication of the current operating regime of the system. The smooth variable structure filter (SVSF) is a relatively new estimation method based on the sliding mode concept, formulated in a predictor-corrector form. For systems with modeling uncertainties, the SVSF has shown to be more accurate and robust when compared with other methods such as the extended Kalman filter (EKF). A newer form of the SVSF makes use of a time-varying smoothing boundary layer (SVSF-VBL). This paper introduces new model-based estimators; based on the IMM strategy combined with the SVSF and SVSF-VBL, referred to as the IMM-SVSF and IMM-SVSF-VBL, respectively. The new strategies are applied to a type of aerospace actuator referred to as an electrohydrostatic actuator, which provides a comprehensive system for fault detection and diagnosis. The results are compared with the popular IMM-EKF strategy.

Invariant Observer Design for a Helicopter UAV Aided Inertial Navigation System

Abstract: The invariant observer is a recently introduced constructive nonlinear design method for symmetry-possessing systems such as the magnetometer-plus-global positioning system (GPS)-aided inertial navigation system (INS) example considered in this paper. The resulting observer guarantees a simplified form of the nonlinear estimation error dynamics, which can be stabilized by a proper choice of observer gains using a nonlinear analysis. A systematic approach to this step is the invariant Extended Kalman Filter (EKF), which is modified from its originally proposed form and applied to the aided INS example to obtain the observer gains. The resulting invariant observer is implemented onboard an outdoor helicopter unmanned aerial vehicle platform and successfully validated in experiment and demonstrates an improvement in performance over a conventional (non-invariant) EKF design.

Adaptive ensemble Kalman filtering of non-linear systems

Abstract: A necessary ingredient of an ensemble Kalman filter (EnKF) is covariance inflation, used to control filter divergence and compensate for model error There is an on-going search for inflation tunings that can be learned adaptively Early in the development of Kalman filtering, Mehra (1970, 1972) enabled adaptivity in the context of linear dynamics with white noise model errors by showing how to estimate the model error and observation covariances We propose an adaptive scheme, based on lifting Mehra’s idea to the non-linear case, that recovers the model error and observation noise covariances in simple cases, and in more complicated cases, results in a natural additive inflation that improves state estimation It can be incorporated into nonlinear filters such as the extended Kalman filter (EKF), the EnKF and their localised versions We test the adaptive EnKF on a 40-dimensional Lorenz96 model and show the significant improvements in state estimation that are possible We also discuss the extent to which such an adaptive filter can compensate for model error, and demonstrate the use of localisation to reduce ensemble sizes for large problems Keywords: ensemble Kalman filter, data assimilation, non-linear dynamics, covariance inflation, adaptive filtering (Published: 2 July 2013) Citation: Tellus A 2013, 65 , 20331, http://dxdoiorg/103402/tellusav65i020331

Discrete Extended Kalman Filter on Lie groups

Abstract: In this paper, we generalize the Discrete Extended Kalman Filter (D-EKF) to the case where the state and the observations evolve on Lie group manifolds. We propose a new filter called Discrete Extended Kalman Filter on Lie Groups (D-LG-EKF). It assumes that the posterior distribution of the state is a concentrated Gaussian distribution on Lie groups. Our formalism yields closed-form equations for both nonlinear discrete propagation and update of the distribution parameters based on the likelihood. We also show that the D-LG-EKF reduces to the traditional D-EKF if the state evolves on an Euclidean space. Our approach leads to a systematic methodology for the design of filters, which is illustrated by the application to a camera pose estimation problem. Results show that the D-LG-EKF outperforms both a constrained D-EKF and a D-EKF applied on the Lie algebra of the Lie group.

Robust Information Fusion Estimator for Multiple Delay-Tolerant Sensors With Different Failure Rates

Abstract: In this paper, the robust information fusion Kalman filtering problem is considered for multi-sensor systems with parameter uncertainties, randomly delayed measurements and sensor failures. The stochastic parameter perturbations are included in the state space models such that the proposed fusion estimator has robustness for the varying system parameters. For each observation subsystem, multiple binary random variables with known probabilities are introduced to model sensor failures and random delays in the measurements. Without resorting to the augmentation of system states and measurements, a robust optimal recursive filter for each subsystem is derived in the linear minimum variance sense by using the innovation analysis method, and the estimation error cross-covariance matrix between any two subsystems is given recursively. Based on the optimal fusion algorithm weighted by matrices, a robust distributed state fusion Kalman filter is derived for the considered system, and the dimension of the designed filter is the same as the original system, which can reduce computation costs as compared with the augmentation method. Moreover, the performance of the designed fusion filter is dependent on the sensor failure rates. Finally, two illustrative examples are given to show the effectiveness of the proposed method.

Kalman Filter-Based Control System for Power Quality Conditioning Devices

Abstract: A new control scheme based on the Kalman filter and the linear quadratic regulator (LQR) is proposed to improve the performance of power quality conditioning devices. Grid perturbations such as load variations, frequency deviation, voltage distortion, line impedance, unbalance, and measurement noise are taken into account. A new model of the plant is developed that allows the use of the LQR controller with a better performance. This new controller structure is feasible thanks to an algorithm based on the Kalman filter (KF), which estimates the state space variables at the point of common coupling, used in the proposed control system. This algorithm is also responsible for generating the references for the controller. The proposed control system was implemented using a digital signal controller. Extensive tests and experimental results are presented in order to verify the performance of the proposal.

Parametric Bayesian Filters for Nonlinear Stochastic Dynamical Systems: A Survey

Abstract: Nonlinear stochastic dynamical systems are commonly used to model physical processes. For linear and Gaussian systems, the Kalman filter is optimal in minimum mean squared error sense. However, for nonlinear or non-Gaussian systems, the estimation of states or parameters is a challenging problem. Furthermore, it is often required to process data online. Therefore, apart from being accurate, the feasible estimation algorithm also needs to be fast. In this paper, we review Bayesian filters that possess the aforementioned properties. Each filter is presented in an easy way to implement algorithmic form. We focus on parametric methods, among which we distinguish three types of filters: filters based on analytical approximations (extended Kalman filter, iterated extended Kalman filter), filters based on statistical approximations (unscented Kalman filter, central difference filter, Gauss-Hermite filter), and filters based on the Gaussian sum approximation (Gaussian sum filter). We discuss each of these filters, and compare them with illustrative examples.

Strapdown inertial navigation system alignment based on marginalised unscented kalman filter

Abstract: This study concerns the strapdown inertial navigation system (SINS) initial alignment under marine mooring condition with large initial error. The ten-dimensional state initial alignment error functions of the SINS with inclusion of non-linear characteristics have been derived. It is pointed out for the first time that the non-linear functions are applied to only a subset of the elements of the state vector, that is, the velocities error and the misalignment angles. Then a computationally efficient refinement of the unscented transformation (UT) called marginalised UT (MUT) is investigated in these special non-linear systems with a linear substructure. A performance comparison between the extended Kalman filter (EKF), the UT-based Kalman filter (UKF) and the MUT-based Kalman filter (MUKF) demonstrates that both the UKF and the MUKF can outperform the EKF and the MUKF and can achieve, if not better, at least a comparable performance to the UKF, at a significantly lower expense.

Minimum-Energy Filtering for Attitude Estimation

Abstract: In this work, we study minimum-energy filtering for attitude kinematics with vectorial measurements using Mortensen's approach. The exact form of a minimum-energy attitude observer is derived and is shown to depend on the Hessian of the value function of an associated optimal control problem. A suitably chosen matrix representation of the Hessian operator leads to a Riccati equation that approximates a minimum-energy attitude filter. An extended version of the proposed approximate filter is included for a situation where there is slowly time-varying bias in the gyro measurements. A unit quaternion version of the proposed filter is derived and shown to outperform the multiplicative extended Kalman filter (MEKF) for situations with large initialization errors or large measurement errors.