Showing papers on "Invariant extended Kalman filter published in 2009"

The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing†

Abstract: The use of heterogeneous, non-collocated measurements for nonlinear structural system identification is explored herein. In particular, this paper considers the example of sensor heterogeneity arising from the fact that both acceleration and displacement are measured at various locations of the structural system. The availability of non-collocated data might often arise in the identification of systems where the displacement data may be provided through global positioning systems (GPS). The well-known extended Kalman filter (EKF) is often used to deal with nonlinear system identification. However, as suggested in (J. Eng. Mech. 1999; 125(2):133–142), the EKF is not effective in the case of highly nonlinear problems. Instead, two techniques are examined herein, the unscented Kalman filter method (UKF), proposed by Julier and Uhlman, and the particle filter method, also known as sequential Monte Carlo method (SMC). The two methods are compared and their efficiency is evaluated through the example of a three degree-of-freedom system, involving a Bouc–Wen hysteretic component, where the availability of displacement and acceleration measurements for different DOFs is assumed. Copyright © 2008 John Wiley & Sons, Ltd.

Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem

Abstract: A new version of the Extended Kalman Filter (EKF) is proposed for nonlinear systems possessing symmetries. Instead of using a linear correction term based on a linear output error, it uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from of a linear state error, but from an invariant state error. The benefit is that the gain and covariance equations converge to constant values on a much bigger set of trajectories than equilibrium points as is the case for the EKF, which should result in a better convergence of the estimation. This filter is applied to the practically relevant problem of estimating the velocity and attitude of a moving rigid body, e.g. an aircraft, from GPS velocity, inertial and magnetic measurements. In this context it can be seen as an extension of the “Multiplicative EKF” often used for quaternion estimation.

Kalman Filtering with Nonlinear State Constraints

Abstract: An analytic method was developed by D. Simon and T. L. Chia to incorporate linear state equality constraints into the Kalman filter. When the state constraint was nonlinear, linearization was employed to obtain an approximately linear constraint around the current state estimate. This linearized constrained Kalman filter is subject to approximation errors and may suffer from a lack of convergence. We present a method that allows exact use of second-order nonlinear state constraints. It is based on a computational algorithm that iteratively finds the Lagrangian multiplier for the nonlinear constraints. Computer simulation results are presented to illustrate the algorithm.

Kalman filtering for positioning and heading control of ships and offshore rigs

Abstract: In this article, we have described the main components of a ship motion-control system and two particular motion-control problems that require wave filtering, namely, dynamic positioning and heading autopilot. Then, we discussed the models commonly used for vessel response and showed how these models are used for Kalman filter design. We also briefly discussed parameter and noise covariance estimation, which are used for filter tuning. To illustrate the performance, a case study based on numerical simulations for a ship autopilot was considered. The material discussed in this article conforms to modern commercially available ship motion-control systems. Most of the vessels operating in the offshore industry worldwide use Kalman filters for velocity estimation and wave filtering. Thus, the article provides an up-to-date tutorial and overview of Kalman-filter-based wave filtering.

Large sample asymptotics for the ensemble Kalman filter

Abstract: The ensemble Kalman filter (EnKF) has been proposed as a Monte Carlo, derivative-free, alternative to the extended Kalman filter, and is now widely used in sequential data assimilation, where state vectors of huge dimension (e.g. resulting from the discretization of pressure and velocity fields over a continent, as considered in meteorology) should be estimated from noisy measurements (e.g. collected at sparse in-situ stations). Even if the state and measurement equations are linear with additive Gaussian white noise, computing and storing the error covariance matrices involved in the Kalman filter is practically impossible, and it has been proposed to represent the filtering distribution with a sample (ensemble) of a few elements and to think of the corresponding empirical covariance matrix as an approximation of the intractable error covariance matrix. Extensions to nonlinear state equations have also been proposed. Surprisingly, very little is known about the asymptotic behaviour of the EnKF, whereas on the other hand, the asymptotic behaviour of many different classes of particle filters is well understood, as the number of particles goes to infinity. Interpreting the ensemble elements as a population of particles with mean-field interactions (and not merely as an instrumental device producing the ensemble mean value as an estimate of the hidden state), we prove the convergence of the EnKF, with the classical rate 1/\sqrt{N}, as the number N of ensemble elements increases to infinity. In the linear case, the limit of the empirical distribution of the ensemble elements is the usual (Gaussian distribution associated with the) Kalman filter, as expected, but in the more general case of a nonlinear state equation with linear observations, this limit differs from the usual Bayesian filter. To get the correct limit in this case, the mechanism that generates the elements in the EnKF should be interpreted as a proposal importance distribution, and appropriate importance weights should be assigned to the ensemble elements.

Adaptive Two-Stage Extended Kalman Filter for a Fault-Tolerant INS-GPS Loosely Coupled System

Abstract: The well-known conventional Kalman filter requires an accurate system model and exact stochastic information. But in a number of situations, the system model has an unknown bias, which may degrade the performance of the Kalman filter or may cause the filter to diverge. The effect of the unknown bias may be more pronounced on the extended Kalman filter (EKF), which is a nonlinear filter. The two-stage extended Kalman filter (TEKF) with respect to this problem has been receiving considerable attention for a long time. Recently, the optimal two-stage Kalman filter (TKF) for linear stochastic systems with a constant bias or a random bias has been proposed by several researchers. A TEKF can also be similarly derived as the optimal TKF. In the case of a random bias, the TEKF assumes that the information of a random bias is known. But the information of a random bias is unknown or partially known in general. To solve this problem, this paper proposes an adaptive two-stage extended Kalman filter (ATEKF) using an adaptive fading EKF. To verify the performance of the proposed ATEKF, the ATEKF is applied to the INS-GPS (inertial navigation system-Global Positioning System) loosely coupled system with an unknown fault bias. The proposed ATEKF tracked/estimated the unknown bias effectively although the information about the random bias was unknown.

The ensemble Kalman filter for combined state and parameter estimation: MONTE CARLO TECHNIQUES FOR DATA ASSIMILATION IN LARGE SYSTEMS

Abstract: he ensemble Kalman filter (EnKF) [1] is a sequential Monte Carlo method that provides an alternative to the traditional Kalman filter (KF) [2], [3] and adjoint or four-dimensional variational (4DVAR) methods [4]-[6] to better handle large state spaces and nonlinear error evolution. EnKF provides a simple conceptual formulation and ease of implementation, since.

On the Convergence of the Ensemble Kalman Filter

Abstract: Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and $L^p$ bounds on the ensemble then give $L^p$ convergence.

Consideration of time-correlated errors in a Kalman filter applicable to GNSS

Abstract: An algorithm for considering time-correlated errors in a Kalman filter is presented. The algorithm differs from previous implementations in that it does not suffer from numerical problems; does not contain inherent time latency or require reinterpretation of Kalman filter parameters, and gives full consideration to additive white noise that is often still present but ignored in previous implementations. Simulation results indicate that the application of the new algorithm yields more realistic and therefore useful state and covariance information than the standard implementation. Results from a field test of the algorithm applied to the problem of kinematic differential GPS demonstrate that the algorithm provides slightly pessimistic covariance estimates whereas the standard Kalman filter provides optimistic covariance estimates.

Unscented Kalman Filters for Multiple Target Tracking With Symmetric Measurement Equations

Abstract: The symmetric measurement equation approach to multiple target tracking is revisited using the unscented Kalman filter. The performance of this filter is compared to the original symmetric measurement equation implementation using an extended Kalman filter. Counterintuitive results are presented and explained for two sets of symmetric measurement equations. We find that the performance of the SME approach is dependent on the interaction of the SME equations and filter used. Furthermore, an SME/unscented Kalman filter pairing is shown to have improved performance versus previous approaches while possessing simpler implementation and equivalent computational complexity.

A DCM Based Orientation Estimation Algorithm with an Inertial Measurement Unit and a Magnetic Compass

Abstract: In this paper, Direction Cosine Matrix (DCM) method for attitude and orientation estimation is discussed. DCM method was chosen due to some advantages over the popular methods such as namely Euler Angle, Quaternion in light of reliability, accuracy and computational efforts. Proposed model for each method is developed for methodology comparison. It is shown that normal Kalman Filter in DCM method is better than extended Kalman Filter in Euler and Quaternion based method because it helps avoid the first order approximation error. Methodology errors are verified using Aerospace Blockset of Matlab Simulink.

Particle Filtering for State Estimation in Nonlinear Industrial Systems

Abstract: State estimation is a major problem in industrial systems, particularly in industrial robotics. To this end, Gaussian and nonparametric filters have been developed. In this paper, the extended Kalman filter, which assumes Gaussian measurement noise, is compared with the particle filter, which does not make any assumption on the measurement noise distribution. As a case study, the estimation of the state vector of an industrial robot is used when measurements are available from an accelerometer that was mounted on the end effector of the robotic manipulator and from the encoders of the joints' motors. It is shown that, in this kind of sensor fusion problem, the particle filter outperforms the extended Kalman filter, at the cost of more demanding computations.

Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Nonlinear Systems

Abstract: Performance of an advanced, derivativeless, sigma-point Kalman filter (SPKF) data assimilation scheme in a strongly nonlinear dynamical model is investigated. The SPKF data assimilation scheme is compared against standard Kalman filters such as the extended Kalman filter (EKF) and ensemble Kalman filter (EnKF) schemes. Three particular cases—namely, the state, parameter, and joint estimation of states and parameters from a set of discontinuous noisy observations—are studied. The problems associated with the use of tangent linear model (TLM) or Jacobian when using standard Kalman filters are eliminated when using SPKF data assimilation algorithms. Further, the constraints and issues of SPKF data assimilation in real ocean or atmospheric models are emphasized. A reduced sigma-point subspace model is proposed and investigated for higher-dimensional systems. A low-dimensional Lorenz 1963 model and a higher-dimensional Lorenz 1995 model are used as the test beds for data assimilation experiments. The ...

Carrier phase-based integrity monitoring for high-accuracy positioning

Abstract: Pseudorange-based integrity monitoring, for example receiver autonomous integrity monitoring (RAIM), has been investigated for many years and is used in various applications such as non-precision approach phase of flight. However, for high-accuracy applications, carrier phase-based RAIM (CRAIM), an extension of pseudorange-based RAIM (PRAIM) must be used. Existing CRAIM algorithms are a direct extension of PRAIM in which the carrier phase ambiguities are estimated together with the estimation of the position solution. The main issues with the existing algorithms are reliability and robustness, which are dominated by the correctness of the ambiguity resolution, ambiguity validation and error sources such as multipath, cycle slips and noise correlation. This paper proposes a new carrier phase-based integrity monitoring algorithm for high-accuracy positioning, using a Kalman filter. The ambiguities are estimated together with other states in the Kalman filter. The double differenced pseudorange, widelane and carrier phase observations are used as measurements in the Kalman filter. This configuration makes the positioning solution both robust and reliable. The integrity monitoring is based on a number of test statistics and error propagation for the determination of the protection levels. The measurement noise and covariance matrices in the Kalman filter are used to account for the correlation due to differencing of measurements and in the construction of the test statistics. The coefficient used to project the test statistic to the position domain is derived and the synthesis of correlated noise errors is used to determine the protection level. Results from four cases based on limited real data injected with simulated cycle slips show that residual cycle slips have a negative impact on positioning accuracy and that the integrity monitoring algorithm proposed can be effective in detecting and isolating such occurrences if their effects violate the integrity requirements. The CRAIM algorithm proposed is suitable for use within Kalman filter-based integrated navigation systems.

Particle and Kalman filtering for state estimation and control of DC motors.

Abstract: State estimation is a major problem in industrial systems. To this end, Gaussian and nonparametric filters have been developed. In this paper the Kalman Filter, which assumes Gaussian measurement noise, is compared to the Particle Filter, which does not make any assumption on the measurement noise distribution. As a case study the estimation of the state vector of a DC motor is used. The reconstructed state vector is used in a feedback control loop to generate the control input of the DC motor. In simulation tests it was observed that for a large number of particles the Particle Filter could succeed in accurately estimating the motor's state vector, but at the same time it required higher computational effort.

The global covariance matrix of tracks fitted with a Kalman filter and an application in detector alignment

Abstract: We present an expression for the covariance matrix of the set of state vectors describing a track fitted with a Kalman filter. We demonstrate that this expression facilitates the use of a Kalman filter track model in a minimum χ 2 algorithm for the alignment of tracking detectors. We also show that it allows to incorporate vertex constraints in such a procedure without refitting the tracks.

Block Kalman Filtering for Large-Scale DSGE Models

Abstract: In this paper block Kalman filters for Dynamic Stochastic General Equilibrium models are presented and evaluated. Our approach is based on the simple idea of writing down the Kalman filter recursions on block form and appropriately sequencing the operations of the prediction step of the algorithm. It is argued that block filtering is the only viable serial algorithmic approach to significantly reduce Kalman filtering time in the context of large DSGE models. For the largest model we evaluate the block filter reduces the computation time by roughly a factor 2. Block filtering compares favourably with the more general method for faster Kalman filtering outlined by Koopman and Durbin (J Time Ser Anal 21:281---296, 2000) and, furthermore, the two approaches are largely complementary.

Self-Tuning Multisensor Weighted Measurement Fusion Kalman Filter

Abstract: For the multisensor systems with unknown noise variances, based on the solution of the matrix equations for the correlation function, the on-line estimators of the noise variance matrices are obtained, whose consistency is proved using the ergodicity of sampled correlation function. Further, two self-tuning weighted measurement fusion Kalman filters are presented for the multisensor systems with identical and different measurement matrices, respectively. Based on the stability of the dynamic error system, a new convergence analysis tool is presented for a self-tuning fuser, which is called the dynamic error system analysis (DESA) method. A new concept of convergence in a realization is presented, which is weaker than the convergence with probability one. It is rigorously proved that the proposed self-tuning Kalman fusers converge to the steady-state optimal Kalman fusers in a realization or with probability one, so that they have asymptotic global optimality. A simulation example for a target tracking system with 3 sensors shows their effectiveness.

Comparison of several space target tracking filters

Abstract: In this paper, we present a comparative study of several nonlinear filters, namely, extended Kalman Filter (EKF), unscented KF (UKF), particle filter (PF), and recursive linear minimum mean square error (LMMSE) filter for the problem of satellite trajectory estimation. We evaluate the tracking accuracy of the above filtering algorithms and obtain the posterior Cramer-Rao lower bound (PCRLB) of the tracking error for performance comparison. Based on the simulation results, we provide recommendations on the practical tracking filter selection and guidelines for the design of observer configurations.

A Brief Tutorial on the Ensemble Kalman Filter

Abstract: The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting. EnKF is related to the particle filter (in this context, a particle is the same thing as an ensemble member) but the EnKF makes the assumption that all probability distributions involved are Gaussian. This article briefly describes the derivation and practical implementation of the basic version of EnKF, and reviews several extensions.