Showing papers on "Invariant extended Kalman filter published in 2010"

Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

Abstract: The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.

Stochastic Stability of the Extended Kalman Filter With Intermittent Observations

Abstract: In this technical note, we analyze the error behavior of the discrete-time extended Kalman filter for nonlinear systems with intermittent observations. Modelling the arrival of the observations as a random process, we show that, given a certain regularity of the system, the estimation error remains bounded if the noise covariance and the initial estimation error are small enough. We also study the effect of different measurement models on the bounds for the error covariance matrices.

Observability-based Rules for Designing Consistent EKF SLAM Estimators

Abstract: In this work, we study the inconsistency problem of extended Kalman filter (EKF)-based simultaneous localization and mapping (SLAM) from the perspective of observability. We analytically prove that when the Jacobians of the process and measurement models are evaluated at the latest state estimates during every time step, the linearized error-state system employed in the EKF has an observable subspace of dimension higher than that of the actual, non-linear, SLAM system. As a result, the covariance estimates of the EKF undergo reduction in directions of the state space where no information is available, which is a primary cause of the inconsistency. Based on these theoretical results, we propose a general framework for improving the consistency of EKF-based SLAM. In this framework, the EKF linearization points are selected in a way that ensures that the resulting linearized system model has an observable subspace of appropriate dimension. We describe two algorithms that are instances of this paradigm. In the first, termed observability constrained (OC)-EKF, the linearization points are selected so as to minimize their expected errors (i.e. the difference between the linearization point and the true state) under the observability constraints. In the second, the filter Jacobians are calculated using the first-ever available estimates for all state variables. This latter approach is termed first-estimates Jacobian (FEJ)-EKF. The proposed algorithms have been tested both in simulation and experimentally, and are shown to significantly outperform the standard EKF both in terms of accuracy and consistency.

A Gaussian mixture PHD filter for extended target tracking

Abstract: In extended target tracking, targets potentially produce more than one measurement per time step. Multiple extended targets are therefore usually hard to track, due to the resulting complex data association. The main contribution of this paper is the implementation of a Probability Hypothesis Density (PHD) filter for tracking of multiple extended targets. A general modification of the PHD filter to handle extended targets has been presented recently by Mahler, and the novelty in this work lies in the realisation of a Gaussian mixture PHD filter for extended targets. Furthermore, we propose a method to easily partition the measurements into a number of subsets, each of which is supposed to contain measurements that all stem from the same source. The method is illustrated in simulation examples, and the advantage of the implemented extended target PHD filter is shown in a comparison with a standard PHD filter.

Constrained Kalman filtering via density function truncation for turbofan engine health estimation

Abstract: Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This article develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter truncates the probability density function (PDF) of the Kalman filter estimate at the known constraints and then computes the constrained filter estimate as the mean of the truncated PDF. The incorporation of state variable constraints increases the computational effort of the filter but also improves its estimation accuracy. The improvement is demonstrated via simulation results obtained from a turbofan engine model. It is also shown that the truncated Kalman filter may provide a more accurate way of incorporating inequality constraints than other constrained filters (e.g. the projection approach to constrained filtering).

Robust Extended Kalman Filtering for Nonlinear Systems With Stochastic Uncertainties

Abstract: In this correspondence paper, a novel robust extended Kalman filter (REKF) for discrete-time nonlinear systems with stochastic uncertainties is proposed. The filter is derived to guarantee an optimized upper bound on the state estimation error covariance despite the model uncertainties as well as the linearization errors. Further analysis shows that the proposed filter has robustness against process noises, measurement noises, and model uncertainties. In addition, the new method is applied in an X-ray pulsar positioning system. It is illustrated through numerical simulations that the REKF is more effective than the standard extended Kalman filter and the extended robust H? filter.

Extended Real Model of Kalman Filter for Time-Varying Harmonics Estimation

Abstract: The effective harmonics estimation for measuring power signals has become an important issue in the power quality assessment. By reviewing those commonly used Kalman filter-based models, some limitations for harmonics estimation can be observed. In this paper an extended real model of Kalman filter combined with a resetting mechanism for accurately tracking time-varying harmonic components of power signals is presented. The usefulness of the proposed algorithm is demonstrated by a simple laboratory setup with LabVIEW program and the dedicated hardware for harmonics monitoring. Results show that the proposed method can achieve more accurate and robust measurement of harmonic amplitudes and phase angles for the time-varying power signals among compared methods while the uncertainty testing performances required by IEC standard 61000-4-30 are satisfied.

The Kalman Filter

Abstract: Although the Kalman filter was orginally developed for space satelilte navigation systems, it proved to be very useful for a host of applications, such as radio communication signals, uninterrupted power supplies, and economics. The Kalman filter is an efficient recursive filter that estimates the internal state of a linear dynamic system from a series of noisy measurements. It is used in a wide range of engineering and econometric applications from radar and computer vision to estimation of structural macroeconomic models [10], and is an important topic in control theory and control systems engineering.

Predictive Iterated Kalman Filter for INS/GPS Integration and Its Application to SAR Motion Compensation

Abstract: This paper deals with the problem of state estimation for the integration of an inertial navigation system (INS) and Global Positioning System (GPS). For a nonlinear system that has the model error and white Gaussian noise, a predictive filter (PF) is used to estimate the model error, and based on this, a modified iterated extended Kalman filter (IEKF) is proposed and is called predictive iterated Kalman filter (PIKF). The basic idea of the PIKF is to compensate the state estimate by the estimated model error. An INS/GPS integration system is implemented using the PIKF and applied to synthetic aperture radar (SAR) motion compensation. Through flight tests, it is shown that the PIKF has an obvious accuracy advantage over the IEKF and unscented Kalman filter (UKF) in velocity.

Data assimilation with the weighted ensemble Kalman filter

Abstract: In this paper, two data assimilation methods based on sequential Monte Carlo sampling are studied and compared: the ensemble Kalman filter and the particle filter. Each of these techniques has its own advantages and drawbacks. In this work, we try to get the best of each method by combining them. The proposed algorithm, called the weighted ensemble Kalman filter, consists to rely on the Ensemble Kalman Filter updates of samples in order to define a proposal distribution for the particle filter that depends on the history of measurement. The corresponding particle filter reveals to be efficient with a small number of samples and does not rely anymore on the Gaussian approximations of the ensemble Kalman filter. The efficiency of the new algorithm is demonstrated both in terms of accuracy and computational load. This latter aspect is of the utmost importance in meteorology or in oceanography since in these domains, data assimilation processes involve a huge number of state variables driven by highly non-linear dynamical models. Numerical experiments have been performed on different dynamical scenarios. The performances of the proposed technique have been compared to the ensemble Kalman filter procedure, which has demonstrated to provide meaningful results in geophysical sciences.

A Non-Gaussian Ensemble Filter Update for Data Assimilation

Abstract: A deterministic square root ensemble Kalman filter and a stochastic perturbed observation ensemble Kalman filter are used for data assimilation in both linear and nonlinear single variable dynamical systems. For the linear system, the deterministic filter is simply a method for computing the Kalman filter and is optimal while the stochastic filter has suboptimal performance due to sampling error. For the nonlinear system, the deterministic filter has increasing error as ensemble size increases because all ensemble members but one become tightly clustered. In this case, the stochastic filter performs better for sufficiently large ensembles. A new method for computing ensemble increments in observation space is proposed that does not suffer from the pathological behavior of the deterministic filter while avoiding much of the sampling error of the stochastic filter. This filter uses the order statistics of the prior observation space ensemble to create an approximate continuous prior probability dis...

Maneuvering Target Tracking in the Presence of Glint using the Nonlinear Gaussian Mixture Kalman Filter

Abstract: The problem of maneuvering target tracking in the presence of glint noise is addressed in this work. The main challenge in this problem stems from its nonlinearity and non-Gaussianity. A new estimator, named as nonlinear Gaussian mixture Kalman filter (NL-GMKF) is derived based on the minimum-mean-square error (MMSE) criterion and applied to the problem of maneuvering target tracking in the presence of glint. The tracking performance of the NL-GMKF is evaluated and compared with the interacting multiple modeling (IMM) implemented with extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF) and the Gaussian sum PF (GSPF). It is shown that the NL-GMKF outperforms these algorithms in several examples with maneuvering target and/or glint noise measurements.

An introduction to Gaussian processes for the Kalman filter expert

Abstract: We examine the close relationship between Gaussian processes and the Kalman filter and show how Gaussian processes can be interpreted using familiar Kalman filter mathematical concepts. We use this insight to develop a novel hybrid filter, which we call the KFGP, for spatial-temporal modelling. The KFGP uses Gaussian process kernels to model the spatial field while exploiting efficient Kalman filter state-based approaches to model the temporal component. We also develop a Gaussian process kernel for the familiar Kalman filter near constant acceleration model.

Catastrophic filter divergence in filtering nonlinear dissipative systems

Abstract: Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demon- strate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catas- trophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations. With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy (A. Majda and M. Grote, Proceedings of the National Academy of Sciences, 104, 1124-1129, 2007), (E. Castronovo, J. Harlim and A. Majda, J. Comput. Phys., 227(7), 3678-3714, 2008), (J. Harlim and A. Majda, J. Comput. Phys., 227(10), 5304-5341, 2008) is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.

Nonlinear Kalman Filtering for Force-Controlled Robot Tasks

Abstract: Introduction- Literature Survey: Autonomous Compliant Motion- Literature Survey: Bayesian Probability Theory- Kalman Filters for Nonlinear Systems- The Non-Minimal State Kalman Filter- Contact Modelling- Geometrical Parameter Estimation and CF Recognition- Experiment: A Cube-In-Corner Assembly- Task Planning with Active Sensing- General Conclusions

Computationally efficient solutions for tracking people with a mobile robot: an experimental evaluation of Bayesian filters

Abstract: Modern service robots will soon become an essential part of modern society. As they have to move and act in human environments, it is essential for them to be provided with a fast and reliable tracking system that localizes people in the neighborhood. It is therefore important to select the most appropriate filter to estimate the position of these persons. This paper presents three efficient implementations of multisensor-human tracking based on different Bayesian estimators: Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Sampling Importance Resampling (SIR) particle filter. The system implemented on a mobile robot is explained, introducing the methods used to detect and estimate the position of multiple people. Then, the solutions based on the three filters are discussed in detail. Several real experiments are conducted to evaluate their performance, which is compared in terms of accuracy, robustness and execution time of the estimation. The results show that a solution based on the UKF can perform as good as particle filters and can be often a better choice when computational efficiency is a key issue.

Comparison of Ensemble Kalman Filters under Non-Gaussianity

Abstract: Recently various versions of ensemble Kalman filters (EnKFs) have been proposed and studied. This work concerns, in a mathematically rigorous manner, the relative performance of two major versions of EnKF when the forecast ensemble is non-Gaussian. The approach is based on the stability of the filtering methods against small model violations, using the expected squared L2 distance as a measure of the deviation between the updated distributions. Analytical and experimental results suggest that both stochastic and deterministic EnKFs are sensitive to the violation of the Gaussian assumption, while the stochastic filter is relatively more stable than the deterministic filter under certain circumstances, especially when there are wild outliers. These results not only agree with previous empirical studies, but also suggest a natural choice of a free parameter in the square root Kalman filter algorithm.

Generalized Multiplicative Extended Kalman Filter for Aided Attitude and Heading Reference System

Abstract: In this paper, we propose a “Generalized Multiplicative Extended Kalman Filter” (GMEKF) to estimate the position and velocity vectors and the orientation of a flying rigid body, using measurements from lowcost Earth-fixed position and velocity, inertial and magnetic sensors. Thanks to well-chosen state and output errors, the gains and covariance equations converge to constant values on a much bigger set of trajectories than equilibrium points as it is the case for the standard Multiplicative Extended Kalman Filter (MEKF). We recover thus the fundamental properties of the Kalman filter in the linear case, especially the convergence and optimality properties, for a large set of trajectories, and it should result in a better convergence of the estimation. We illustrate the good performance and the nice properties of the GMEKF on simulation and on experimental comparisons with a commercial system.

Nonlinear Estimation With State-Dependent Gaussian Observation Noise

Abstract: We consider the problem of estimating the state of a system when measurement noise is a function of the system's state. We propose generalizations of the extended Kalman filter and the iterated extended Kalman filter that can be utilized when the state estimate distribution is approximately Gaussian. The state estimate is computed by an iterative root-searching method that maximizes a maximum likelihood function. The new filter allows for the consistent treatment of a class of control problem involving nonlinear estimation from measurements with state-dependent noise. The effectiveness of the estimation algorithm is illustrated for a control problem with a mobile bearing-only sensor.

Particle Filtering for Attitude Estimation Using a Minimal Local-Error Representation

Abstract: A computationally efficient spacecraft attitude estimation particle filter is derived. The quaternion is used to represent the global attitude and the modified Rodrigues parameters are used to represent the local errors. The mean attitude is computed through the weighted sum of the modified Rodrigues parameter particles. The idea of progressive correction is employed to ensure the proper operation of the particle filter. Simulation results indicate that the performance of the particle filter exceeds that of the extended Kalman filter or the unscented Kalman filter for very large initialization errors.

Performance Comparison of Different Forms of Kalman Filter Approaches for a Vector‐Based GNSS Signal Tracking Loop

Abstract: In this paper we discuss several different Kalman filter methods to estimate signal parameters in a GNSS receiver's signal tracking loop with a special emphasis given on how to construct the measurement residuals and the relevant design matrix There are three variations of error state Kalman filters for a loop filter, and a direct state Kalman filter for the entire signal tracking loop Based on detailed signal models for a signal tracking loop, four different Kalman filters for signal tracking are compared with the traditional one Block diagram implementation of the direct state Kalman filter for the entire signal tracking loop is proposed by using the analogy relationship with the traditional method in a scalar tracking mode and a vector tracking mode Numerical simulation results show that the Kalman filter methods provide 2 to 3 dB carrier-to-noise ratio gain in carrier phase and Doppler tracking compared to the traditional method

On the Consider Kalman Filter

Abstract: Parameter errors in dynamic and measurement models of dynamic systems can result in poor state estimates when using a traditional Kalman fllter structure. In dealing with these parameter errors it is possible to: 1) Ignore them completely; 2) Add the parameters as additional states to be estimated; or 3) \Consider" the error in the state covariance matrix by introducing additional parameter covariance matrices. This paper analyzes the efiect of using all three of these types of fllters on a simple asteroid rendezvous scenario to determine the applicability of each. Two types of consider Kalman fllters are explored, namely an Augmented Measurement Consider Kalman Filter and a Minimum Variance Consider Kalman Filter. This paper flnds that a Minimum Variance Consider Kalman Filter can provide not only improved state estimates to a traditional Kalman fllter, but also produces consistent results from a statistical perspective.

Distributed nonlinear Kalman filtering with applications to wireless localization

Abstract: We study the problem of distributed state-space estimation, where a set of nodes are required to estimate the state of a nonlinear state-space system based on their observations. We extend our previous work on distributed Kalman filtering to the nonlinear case, and propose algorithms for Extended and Unscented Kalman filtering. The resulting algorithms are robust to node and link failure, scalable, and fully distributed, in the sense that no fusion center is required, and nodes communicate with their neighbors only. We apply the algorithms to the problem of estimating the position of every node in an ad-hoc network, also known as wireless localization. Simulation results illustrate the performance of the proposed algorithms.

Constrained Nonlinear State Estimation Using Ensemble Kalman Filters

Abstract: Recursive estimation of states of constrained nonlinear dynamic systems has attracted the attention of many researchers in recent years. In this work, we propose a constrained recursive formulation...

On the optimality of two‐stage Kalman filtering for systems with unknown inputs

Abstract: This paper is concerned with the optimal solution of two-stage Kalman filtering for linear discrete-time stochastic time-varying systems with unknown inputs affecting both the system state and the outputs. By means of a newly-presented modified unbiased minimum-variance filter (MUMVF), which appears to be the optimal solution to the addressed problem, the optimality of two-stage Kalman filtering for systems with unknown inputs is defined and explored. Two extended versions of the previously proposed robust two-stage Kalman filter (RTSKF), augmented-unknown-input RTSKF (ARTSKF) and decoupled-unknown-input RTSKF (DRTSKF), are presented to solve the general unknown input filtering problem. It is shown that under less restricted conditions, the proposed ARTSKF and DRTSKF are equivalent to the corresponding MUMVFs. An example is given to illustrate the usefulness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Ocean Vessel Trajectory Estimation and Prediction Based on Extended Kalman Filter

Abstract: The accurate estimation and prediction of the trajectories of maneuvering vessels in ocean navigation are important tools to improve maritime safety and security. Therefore, many conventional ocean navigation systems and Vessel Traffic Management & Reporting Services are equipped with Radar facilities for this purpose. However, the accuracy of the predictions of maneuvering trajectories of vessels depends mainly on the goodness of estimation of vessel position, velocity and acceleration. Hence, this study presents a maneuvering ocean vessel model based on a curvilinear motion model with the measurements based on a linear position model for the same purpose. Furthermore, the system states and measurements models associated with a white Gaussian noise are also assumed. The Extended Kalman Filter is proposed as an adaptive filter algorithm for the estimation of position, velocity and acceleration that are used for prediction of maneuvering ocean vessel trajectory. Finally, the proposed models are implemented and successful computational results are obtained with respect to prediction of maneuvering trajectories of vessels in ocean navigation in this study. KeywordsTrajectory estimation; Trajectory prediction; Target tracking; Extended Kalman Filter; Curvilinear motion model.

Combination of Kalman Filter and Least-Error Square Techniques in Power System

Abstract: An algorithm based on the concept of combining Kalman filter and least-error square (LES) techniques is proposed in this paper. The algorithm is intended to estimate signal attributes like amplitude, frequency and phase angle in the online mode. This technique can be used in protection relays, digital AVRs, DGs, DSTATCOMs, FACTS, and other power electronics applications. The Kalman filter is modified to operate on a fictitious input signal and provides precise estimation results insensitive to noise and other disturbances. At the same time, the LES system has been arranged to operate in critical transient cases to compensate the delay and inaccuracy identified because of the response of the standard Kalman filter. Practical considerations such as the effect of noise, higher order harmonics, and computational issues of the algorithm are considered and tested in the paper. Several computer simulations and a laboratory test are presented to highlight the usefulness of the proposed method. Simulation results show that the proposed technique can simultaneously estimate the signal attributes, even if it is highly distorted due to the presence of nonlinear loads and noise.

Generalized particle flow for nonlinear filters

Abstract: We generalize the theory of particle flow to stabilize the nonlinear filter. We have invented a new nonlinear filter that is vastly superior to the classic particle filter and the extended Kalman filter (EKF). In particular, the computational complexity of the new filter is many orders of magnitude less than the classic particle filter with optimal estimation accuracy for problems with dimension greater than 4. Our accuracy is typically several orders of magnitude better than the EKF for nonlinear problems. We do not resample, and we do not use any proposal density from an EKF or UKF or other filter. Moreover, our new algorithm is deterministic, and we do not use any MCMC methods; this is a radical departure from other particle filters. The new filter implements Bayes' rule using particle flow rather than with a pointwise multiplication of two functions; this avoids one of the fundamental and well known problems in particle filters, namely "particle degeneracy." In addition, we explicitly stabilize our particle filter using negative feedback, unlike standard particle filters, which are generally very inaccurate for plants with slow mixing or unstable dynamics. This stabilization improves performance by several orders of magnitude for difficult problems.