Showing papers on "Invariant extended Kalman filter published in 2001"

The square-root unscented Kalman filter for state and parameter-estimation

Abstract: Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (eg, learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. (1997), proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state estimation. This was extended to parameter estimation by Wan and Van der Merwe et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL/sup 3/)/sup l/ for general state-space problems. When the EKF is applied to parameter estimation, the special form of the state-space equations allows for an O(L/sup 2/) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L/sup 3/) for general state estimation and O(L/sup 2/) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L/sup 3/)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.

An extended Kalman filter for quaternion-based orientation estimation using MARG sensors

Abstract: Presents an extended Kalman filter for real-time estimation of rigid body orientation using the newly developed MARG (magnetic, angular rate, and gravity) sensors. Each MARG sensor contains a three-axis magnetometer, a three-axis angular rate sensor, and a three-axis accelerometer. The filter represents rotations using quaternions rather than Euler angles, which eliminates the long-standing problem of singularities associated with attitude estimation. A process model for rigid body angular motions and angular rate measurements is defined. The process model converts angular rates into quaternion rates, which are integrated to obtain quaternions. The Gauss-Newton iteration algorithm is utilized to find the best quaternion that relates the measured accelerations and earth magnetic field in the body coordinate frame to calculated values in the earth coordinate frame. The best quaternion is used as part of the measurements for the Kalman filter. As a result of this approach, the measurement equations of the Kalman filter become linear, and the computational requirements are significantly reduced, making it possible to estimate orientation in real time. Extensive testing of the filter with synthetic data and actual sensor data proved it to be satisfactory. Test cases included the presence of large initial errors as well as high noise levels. In all cases the filter was able to converge and accurately track rotational motions.

Stochastic Methods for Sequential Data Assimilation in Strongly Nonlinear Systems

Abstract: This paper considers several filtering methods of stochastic nature, based on Monte Carlo drawing, for the sequential data assimilation in nonlinear models. They include some known methods such as the particle filter and the ensemble Kalman filter and some others introduced by the author: the second-order ensemble Kalman filter and the singular extended interpolated filter. The aim is to study their behavior in the simple nonlinear chaotic Lorenz system, in the hope of getting some insight into more complex models. It is seen that these filters perform satisfactory, but the new filters introduced have the advantage of being less costly. This is achieved through the concept of second-order-exact drawing and the selective error correction, parallel to the tangent space of the attractor of the system (which is of low dimension). Also introduced is the use of a forgetting factor, which could enhance significantly the filter stability in this nonlinear context.

Better proposal distributions: object tracking using unscented particle filter

Abstract: Tracking objects involves the modeling of non-linear non-Gaussian systems. On one hand, variants of Kalman filters are limited by their Gaussian assumptions. On the other hand, conventional particle filter, e.g., CONDENSATION, uses transition prior as the proposal distribution. The transition prior does not take into account current observation data, and many particles can therefore be wasted in low likelihood area. To overcome these difficulties, unscented particle filter (UPF) has recently been proposed in the field of filtering theory. In this paper, we introduce the UPF framework into audio and visual tracking. The UPF uses the unscented Kalman filter to generate sophisticated proposal distributions that seamlessly integrate the current observation, thus greatly improving the tracking performance. To evaluate the efficacy of the UPF framework, we apply it in two real-world tracking applications. One is the audio-based speaker localization, and the other is the vision-based human tracking. The experimental results are compared against those of the widely used CONDENSATION approach and have demonstrated superior tracking performance.

Model-based 3D tracking of an articulated hand

Abstract: This paper presents a practical technique for model-based 3D hand tracking. An anatomically accurate hand model is built from truncated quadrics. This allows for the generation of 2D profiles of the model using elegant tools from projective geometry, and for an efficient method to handle self-occlusion. The pose of the hand model is estimated with an Unscented Kalman filter (UKF), which minimizes the geometric error between the profiles and edges extracted from the images. The use of the UKF permits higher frame rates than more sophisticated estimation methods such as particle filtering, whilst providing higher accuracy than the extended Kalman filter The system is easily scalable from single to multiple views, and from rigid to articulated models. First experiments on real data using one and two cameras demonstrate the quality of the proposed method for tracking a 7 DOF hand model.

Variance reduced ensemble Kalman filtering

Abstract: A number of algorithms to solve large-scale Kalman filtering problems have been introduced recently. The ensemble Kalman filter represents the probability density of the state estimate by a finite number of randomly generated system states. Another algorithm uses a singular value decomposition to select the leading eigenvectors of the covariance matrix of the state estimate and to approximate the full covariance matrix by a reduced-rank matrix. Both algorithms, however, still require a huge amount of computer resources. In this paper the authors propose to combine the two algorithms and to use a reduced-rank approximation of the covariance matrix as a variance reductor for the ensemble Kalman filter. If the leading eigenvectors explain most of the variance, which is the case for most applications, the computational burden to solve the filtering problem can be reduced significantly (up to an order of magnitude).

Finite-horizon robust Kalman filter design

Abstract: We study the problem of finite-horizon Kalman filtering for systems involving a norm-bounded uncertain block. A new technique is presented for robust Kalman filter design. This technique involves using multiple scaling parameters that ran be optimized by solving a semidefinite program. The use of optimized scaling parameters leads to an improved design. A recursive design method that can be applied to real-time applications is also proposed.

Efficient derivative-free Kalman filters for online learning.

Abstract: The extended Kalman filter(EKF) is considered one of the most ef- fective methods for both nonlinear state estimation and parameter estimation(e.g., learning the weights of a neural network). Recently, a number of derivative free alternatives to the EKF for state estimation have been proposed. These include the Unscented Kalman Filter(UKF) (1, 2), the Central Difference Filter(CDF) (3) and the closely related Divided Difference Filter(DDF) (4). These filters consistently outperform the EKF for state estimation, at an equal computational complexity of . Extension of the UKF to parameter estimation was presented by Wan and van der Merwe in (5, 6). In this paper, we further develop these techniques for parameter estimation and neural network training. The extension of the CDF and DDF filters to parameter estimation, and their relation to UKF parameter estimation is presented. Most significantly, this paper introduces efficient square-root forms of the different filters. This enables an implementation for parameter esti- mation (equivalent to the EKF), and has the added benefit of improved numerical stability and guaranteed positive semi-definiteness of the Kalman filter covariances.

Optimality of variational data assimilation and its relationship with the Kalman filter and smoother

Abstract: The known properties of equivalence between four-dimensional variational (4D-Var) data assimilation and the Kalman filter as well as the fixed-interval Kalman smoother point to particular optimal properties of 4D-Var. In the linear context, the 4D-Var solution is optimal, not only with respect to the model trajectory segment over the assimilation time interval, but also with respect to any model state at a single observation time level; in the batch processing (cycling 4D-Var) method, the information in 4D-Var is fully transferred from one batch to the next by the background term; 4D-Var allows the processing of observations in subsets, while the final solution is optimal as all observations are processed simultaneously. These properties hold even for models that are imperfect, as well as not invertible. Various properties of equivalence of 4D-Var to the Kalman filter and smoother result from these optimality properties of 4D-Var. Further, we show that the fixed-lag Kalman smoother may also be constructed in an optimal way using a multiple batch-processing 4D-Var approach. While error covariances are crucial for the equivalence, practical techniques for evaluating error covariances in the framework of cycling 4D-Var are discussed.

State Estimation Using a Reduced-Order Kalman Filter

Abstract: Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for a wide class of linear systems, and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time-dependent error covariance matrix, which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model, which suggests the use of reduced-order error models to obtain near-optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced-order approximation of the forecast error system. This reduced-order system is obtained by balanced truncation of the Hankel operator ...

Towards constant time SLAM using postponement

Abstract: Many recent approaches to simultaneous localisation and mapping (SLAM) use an extended Kalman filter (EKF) to update and maintain a map of vehicle location. and multiple feature positions as a sensor moves through a scene. Although it is a highly powerful and well-used tool, it suffers from a well-known complexity problem. In this paper we outline the postponement technique which allows for much greater flexibility on when to use the available processing time, while not affecting the optimality of the filter. It works by updating a constant-sized data set based on current measurements, which can be used to affect the updates on all unobserved parts of the map at a later stage. By expanding the set of updated features when each new feature is observed we show that the full map update can be postponed indefinitely. We also demonstrate how postponement can be used to improve the performance of sub-optimal algorithms by applying it to a simple constant time method.

Improved Kalman filter design for three-dimensional radar tracking

Abstract: The problem of three-dimensional (3D) radar tracking is considered. The usual tracking filter design relying on first-order (or linear) approximations leads to poor convergence and erratic filter behavior in highly nonlinear situations. Simple filter algorithms that can overcome these ill effects are developed for two different types of 3D radar measurement. For each type of radar measurement, an accurate expression for the measurement covariance is obtained by evaluating inherent nonlinearities of radar measurements via coordinate transformation. Then algebraic manipulations and reasonable approximations are employed to yield a simple filter formulation based on the expression. The resulting filter equations are similar to the extended Kalman filter (EKF) and provide some useful insights into the behavior of linearized Kalman filters designed with radar measurements. Finally, simulation results show that the proposed approach is very effective in accounting for the measurement nonlinearities.

Method and apparatus for real-time estimation of physiological parameters

Abstract: A real-time glucose estimator uses a linearized Kalman filter to determine a best estimate of glucose level in real time. The real-time glucose estimator receives at least one measurement corresponding to glucose level. The measurement can be obtained with one or more sensors and is provided to the linearized Kalman filter in real time. The linearized Kalman filter has dynamic models and executes a recursive routine to determine the best estimate of glucose level based upon the measurement. Additional information can be provided to the linearized Kalman filter for initialization, configuration, and the like. Outputs of the linearized Kalman filter can be provided to a patient health monitor for display or for statistical testing to determine status of the real-time glucose estimator. The real-time glucose estimator can be implemented using a software algorithm.

Pseudo-linear estimator for bearings-only passive target tracking

Abstract: The pseudo-linear extended Kalman tracking filter previously developed for passive target tracking using bearings-only measurements required the use of a null target state vector as an initial estimate to obtain convergence for all types of scenario. The pseudo-linear estimator is projected in such a way that it does not require any initial estimate at all and at the same time offers all the features of the extended Kalman filter based pseudo-linear filter; namely sequential processing, flexibility to adopt the variance of each measurement, etc. The algorithm is tested in Monte Carlo simulations and its results are presented for two typical scenarios at various noise levels.

Detection of the EEG Artifacts by the Means of the (Extended) Kalman Filter

Abstract: This paper presents a new approach for detection of artifacts in sleep electroencephalogram (EEG) recordings. The proposed approach is based on Kalman filter. The idea of this approach consist in embedding the AR model into the Kalman Filter which makes possible to use such KF AR (Kalman Filter AR) models for linear prediction of non-stationary signals. Such model can be set up to detect and follow discrete dynamic changes of the signal. For detection of the EEG artifacts we have exploited the evolution of the state noise - increase in state noise indicate the dynamic change of the signal. The evaluation of the results was done by the Receiver-Operator Characteristics (ROC) curves in terms of the specificity and the sensitivity. For 90% of the specificity the best achieved value of the sensitivity using KF AR model was 33%. In order to achieve better results we have tried the following modification: instead of the Kalman Filter we have used extended Kalman Filter and instead of the AR model a neural network. The preliminary results look promissing: for 90% of the specificity we have achieved 65% of the sensitivity.

Gaussian particle filtering

Abstract: Sequential Bayesian estimation for dynamic state space models involves recursive estimation of hidden states based on noisy observations. The update of filtering and predictive densities for nonlinear models with non-Gaussian noise using Monte Carlo particle filtering methods is considered. The Gaussian particle filter (GPF) is introduced, where densities are approximated as a single Gaussian, an assumption which is also made in the extended Kalman filter (EKF). It is analytically shown that, if the Gaussian approximations hold true, the GPF minimizes the mean square error of the estimates asymptotically. The simulations results indicate that the filter has improved performance compared to the EKF, especially for highly nonlinear models where the EKF can diverge.

H/sub /spl infin//-learning of layered neural networks

Abstract: Although the backpropagation (BP) scheme is widely used as a learning algorithm for multilayered neural networks, the learning speed of the BP algorithm to obtain acceptable errors is unsatisfactory in spite of some improvements such as introduction of a momentum factor and an adaptive learning rate in the weight adjustment. To solve this problem, a fast learning algorithm based on the extended Kalman filter (EKF) is presented and fortunately its computational complexity has been reduced by some simplifications. In general, however, the Kalman filtering algorithm is well known to be sensitive to the nature of noises which is generally assumed to be Gaussian. In addition, the H/sub /spl infin// theory suggests that the maximum energy gain of the Kalman algorithm from disturbances to the estimation error has no upper bound. Therefore, the EKF-based learning algorithms should be improved to enhance the robustness to variations in the initial values of link weights and thresholds as well as to the nature of noises. The paper proposes H/sub /spl infin//-learning as a novel learning rule and to derive new globally and locally optimized learning algorithms based on H/sub /spl infin//-learning. Their learning behavior is analyzed from various points of view using computer simulations. The derived algorithms are also compared, in performance and computational cost, with the conventional BP and EKF learning algorithms.

A note on uniform observability

Abstract: We prove that the classical inequality P/sub t//spl les//spl Oscr//sub t//sup -1/C/sub t/ relating the variance of the Kalman filter estimate, the observability matrix, and the controllability matrix is not true. This inequality is the cornerstone of the asymptotic stability theory of the Kalman filter for time-varying systems. We provide another inequality of the same type.

Derivation of a sawtooth iterated extended Kalman smoother via the AECM algorithm

Abstract: The iterated extended Kalman smoother (IEKS) is derived under expectation-maximization (EM) algorithm formalism, providing insight into the behavior of the suboptimal extended Kalman filter (EKF) and smoother (EKS). Through an investigation of smoothing algorithms that result from variants of the EM algorithm, the sawtooth iterated extended Kalman smoother (SIEKS) and its computationally inexpensive counterparts are proposed via the alternating expectation conditional maximization (AECM) algorithm. The SIEKS is guaranteed to produce a sequence estimate that moves up the likelihood surface. Numerical simulations including frequency tracking examples display the superior performance of the sawtooth EKF over the standard EKF for a range of nonlinear signal models.

Sensor fusion based on fuzzy Kalman filter

Abstract: We present a fuzzy Kalman filter, which is based on fuzzy logic theory and a Kalman filter. It is similar to a Kalman filter when a linear system with Gaussian noise is considered. However, when non-Gaussian noise is introduced, it is shown that the fuzzy Kalman filter outperforms the Kalman filter and the Kalman filter does not work well. We demonstrate the performances of the Kalman filter and the fuzzy Kalman filter for a position estimation application under different circumstances. Comparisons are made to draw conclusions.

Nonlinear estimation of aircraft models for on-line control customization

Abstract: This paper describes a new nonlinear estimation procedure used to estimate and track the parameters of a nonlinear aircraft. The Unscented Kalman Filter (UKF) is developed and compared to the more traditional Extended Kalman Filter (EKF). State and parameters are estimated on the F-15 for both a complex maneuver and a maneuver with failure. The algorithms have access to the nonlinear dynamic equations, but not the aircraft engine models, aerodynamic models, or atmospheric models. Parameters describing these unknown dynamics are estimated in the EKF and UKF algorithms. Results show the UKF to be more accurate than the EKF, and track all parameters very well at all times, even after a 50% failure of the stabilator. The aerodynamic forces and moments, while difficult to track immediately after the failure because of the discontinuous nonlinearity, did recover quickly and stay within the predicted bounds.

Improved rotor speed estimation using two Kalman filter-based algorithms

Abstract: This paper proposes the application of two new Kalman filter-based algorithms to realise a speed-sensorless vector control of induction motor drives. The first one is a linear Kalman filter (LKF)-based algorithm that estimates the equivalent disturbance torque, which is compensated by the injection of a feedforward signal. The latter is an extended Kalman filter (EKF)-based algorithm used to obtain a correct implementation of sensorless vector control, since it estimates both the rotor flux components and speed. The mathematical EKF-model is accurate because of the equivalent-disturbance compensation obtained from the LKF-based observer. The rotor speed estimate is very good in the whole velocity range including zero value. The results show the effectiveness of the proposed control scheme.

A blind network of extended Kalman filters for nonstationary channel equalization

Abstract: A blind network of extended Kalman filters (NEKF) is introduced for nonstationary linear channel equalization. The structure of NKF was recently suggested for optimal channel equalization. As the knowledge of the channel is the main constraint within the NKF equalizer, we here propose to extend the state to estimate, that was previously formed by the last M transmitted symbols, to the time-varying channel coefficients. The observation model becomes nonlinear suggesting thus extended Kalman filtering for state estimation. The proposed NEKF algorithm is completely blind towards any learning phase, with fast convergence properties. Compared to the blind Bayesian algorithm proposed by Iltis et al., (1994), the NEKF-based equalizer shows good performance with a much lower complexity.

Development of a fuzzy logic-based adaptive Kalman filter

Abstract: In this paper, after reviewing the traditional Kalman filter formulation, a development of a fuzzy logic-based adaptive Kalman filter is outlined. The adaptation is in the sense of adaptively tuning, on-line, the measurement noise covariance matrix R or the process noise covariance matrix Q. This improves the Kalman filter performance and prevents filter divergence when R or Q are uncertain. Based on the whiteness of the filter innovation sequence and employing the covariance-matching technique the tuning process is carried out by a fuzzy inference system. If a statistical analysis of the innovation sequence shows discrepancies with its expected statistics then a fuzzy inference system adjusts a factor through which the matrices R or Q are tuned on line. This fuzzy logic-based adaptive Kalman filter is tested on a numerical example. The results are compared with these obtained using a conventional Kalman filter and a traditionally adapted Kalman filter. The fuzzy logic-based adaptive Kalman filter showed better results than its traditional counterparts.

Kalman filter with adaptive measurement variance estimator

Abstract: A Kalman filtering technique employing an adaptive measurement variance estimator is disclosed. The Kalman filtering technique includes a signal filtering mechanism, the signal filtering mechanism futher includes a Kalman filter and a variance estimator. The variance estimation used in the filtering includes estimating the variance of the measured quantity signal and generating the variance estimate signal for use in filtering the input signal and the measured quantity signal, wherein estimating the variance of the measured quantity signal includes determining a smoothed estimate of the measured quantity's variance from the measured quantity signal. The invention also manifests itself as a method for filtering and estimating, a program storage medium encoded with instructions that, when executed by a computer, performs such a method, an electronic computing device programmed to perform such a method, and a transmission medium over which the method is performed.

Comparative analysis of backpropagation and extended Kalman filter in pattern and batch forms for training neural networks

Abstract: The extended Kalman filter (EKF) algorithm has been used for training neural networks. Like the backpropagation (BP) algorithm, the EKF algorithm can be in pattern or batch form. But the batch form EKF is different from the gradient averaging in standard batch mode BP. The paper compares backpropagation and extended Kalman filter in pattern and batch forms for neural network trainings. For each comparison between the batch-mode EKF and BP, the same batch data size is used. An overall RMS error computed for all training examples is adopted in the paper for the comparison, which is found to be especially beneficial to pattern mode EKF and BP trainings. Simulation of the network training with different batch data sizes shows that EKF and BP in batch-form usually are more stable and can obtain smaller RMS error than in pattern-form. However, too large batch data size can let the BP trap to a "local minimum", and can also reduces the network training effect of the EKF algorithm.