Sigma Points Research Articles

Tightly-coupled stereo visual-inertial navigation using point and line features.

This paper presents a novel approach for estimating the ego-motion of a vehicle in dynamic and unknown environments using tightly-coupled inertial and visual sensors. To improve the accuracy and robustness, we exploit the combination of point and line features to aid navigation. The mathematical framework is based on trifocal geometry among image triplets, which is simple and unified for point and line features. For the fusion algorithm design, we employ the Extended Kalman Filter (EKF) for error state prediction and covariance propagation, and the Sigma Point Kalman Filter (SPKF) for robust measurement updating in the presence of high nonlinearities. The outdoor and indoor experiments show that the combination of point and line features improves the estimation accuracy and robustness compared to the algorithm using point features alone.

Fault detection and identification of non‐linear hybrid system using self‐switched sigma point filter bank

This study proposes a scheme for fault detection and identification (FDI) of a class of non-linear hybrid systems. For fault detection, the proposed method uses a self-switched unscented Kalman filter (UKF) where a component filter with the appropriate dynamic model is chosen to suit the current mode of the hybrid plant. The mode of the hybrid plant, usually defined by system states, is deduced based on the state estimates. The statistical tests of measured datasets are used to detect and identify the fault parameters. For fault identification, a bank of such switched UKF filters is used. A three-tank system has been used to demonstrate the effectiveness of the scheme. Different types of faults, that is, leakage, clogging and abrupt changes in inflow (actuator fault) have been considered. t-Statistical test has been performed on the residuals for FDI and threshold calculation purposes. It is shown that all the types of faults can be successfully detected and identified by the proposed method. The performance of the proposed system is compared with the same for an extended Kalman filter-based FDI system. It has been shown that the UKF-based scheme has lower latency and higher range coverage.

Algorithm of gaussian sum filter based on high-order UKF for dynamic state estimation

In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. A new version of Gaussian sum estimation algorithm is developed here based on high-order unscented Kalman filter (HUKF). A sigma point selection method, high-order unscented transformation (HUT) technique is proposed for the HUKF, which can approximate the Gaussian distributions more accurately. We present the systematic formulation of Gaussian filters and develop efficient and accurate numerical integration of the optimal filter. We then go on to extend the use of the HUKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. The resulting filtering algorithm, called the Gaussian sum high-order unscented Kalman filter (GS-HUKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. It is corroborated in the theoretical analysis and the simulation that the proposed Gaussian sum HUKF has integrated advantages with respect to computational accuracy and time complexity for nonlinear non-Gaussian filtering problems.

Implicit-Runge–Kutta-based methods for fast, precise, and scalable uncertainty propagation

A large class of methods for propagating the state of an object and its associated uncertainty require the propagation of an ensemble of particles or states through nonlinear dynamics. Existing sigma point- or particle-based methods for uncertainty propagation solve the ensemble of initial value problems one-by-one without exploiting the proximity of the initial conditions. In this paper, we demonstrate how implicit Runge–Kutta methods can be modified to solve initial-value-problem ensembles collectively in order to significantly reduce the computational cost of uncertainty propagation. Examples and variations of this approach are given in the context of the perturbed two-body problem of orbital mechanics that arises in space-object tracking, conjunction analysis, maneuver detection, and other Astrodynamics functions. Particular attention is given to the accuracy of the solution to the initial-value-problem ensemble when the particles are propagated collectively.

State of charge estimation for LiMn2O4 power battery based on strong tracking sigma point Kalman filter

The State of Charge (SOC) estimation is important since it has a crucial role in the operation of Electrical Vehicle (EV) power battery. This paper built an Equivalent Circuit Model (ECM) of the LiMn2O4 power battery, and vast characteristics experiments were undertaken to make the model identification and thus the battery SOC estimation was realized. The SOC estimation was based on the Strong Tracking Sigma Point Kalman Filter (STSPKF) algorithm. The comparison of experimental and simulated results indicates that the STSPKF algorithm performs well in estimating the battery SOC, which has the advantages of tracking the variables in real-time and adjusting the error covariance by taking the Strong Tracking Factor (STF) into account. The results also show that the STSPKF algorithm estimated the SOC more accurately than the Extended Kalman Filter (EKF) algorithm.

Gravity Matching Aided Inertial Navigation Technique Based on Marginal Robust Unscented Kalman Filter

This paper is concerned with the topic of gravity matching aided inertial navigation technology using Kalman filter. The dynamic state space model for Kalman filter is constructed as follows: the error equation of the inertial navigation system is employed as the process equation while the local gravity model based on 9-point surface interpolation is employed as the observation equation. The unscented Kalman filter is employed to address the nonlinearity of the observation equation. The filter is refined in two ways as follows. The marginalization technique is employed to explore the conditionally linear substructure to reduce the computational load; specifically, the number of the needed sigma points is reduced from 15 to 5 after this technique is used. A robust technique based on Chi-square test is employed to make the filter insensitive to the uncertainties in the above constructed observation model. Numerical simulation is carried out, and the efficacy of the proposed method is validated by the simulation results.

Scale-corrected minimal skew simplex sampling UKF for BLDCM sensorless control

In this paper, a scale-corrected minimal skew simplex sampling unscented Kalman filter (UKF) algorithm for the permanent magnet (PM) brushless DC motors (BLDCM) sensorless control has been studied to cancel the position sensor by the use of a systematical and analytical approach. Compared with the general UKF, the sampling method with the least Sigma points called minimal skew simplex sampling is adopted to reduce amount of computation and increase the estimation precision. Moreover, the scale-corrected strategy is introduced into the minimal skew simplex sampling UKF to overcome the nonlocal effects. On the other hand, for more easily calculating the value of back-EMF, the shape function of counter electromotive force is approximated by a series of sine and cosine functions based on the law of Fourier series. The purpose of the problem addressed is, by the method of scale-corrected minimal skew simplex sampling UKF, to properly estimate the rotor speed and position without installing encoders. Finally, the effectiveness of the proposed sensorless control technique is verified by simulation in MATLAB/Simulink.

Economic Multi-stage Output Feedback NMPC using the Unscented Kalman Filter

Nonlinear Model predictive control (NMPC) is a popular control strategy for highly nonlinear chemical processes. The ability to handle safety and environmental constraints along with the use of an economic objective makes NMPC highly appealing to industries. The performance of NMPC depends strongly on the accuracy of the model. In reality, there always are plant-model mismatch and state estimation errors. Hence the NMPC controller must be robust to uncertainties in the model as well as against estimation errors. Among the several approaches presented in the literature, the scenario-tree based multi-stage NMPC approach is a non-conservative and efficient formulation. In this approach, the evolutions of the plant for different realizations of the uncertainties are considered as different scenarios and the optimization problem is formulated as a multi-stage stochastic programming problem with recourse. In this work, we consider multi-stage output feedback NMPC using the Unscented Kalman Filter (UKF) where the nonlinearities are represented using deterministically chosen sigma points for state estimation. In the control problem, we explicitly consider the UKF estimation equations to predict the future evolution of the system. The proposed approach is illustrated by simulation results of fed-batch chemical reactor with an economic cost function.

Model updating with constrained unscented Kalman filter for hybrid testing

The unscented Kalman filter (UKF) has been developed for nonlinear model parametric identification, and it assumes that the model parameters are symmetrically distributed about their mean values without any constrains. However, the parameters in many applications are confined within certain ranges to make sense physically. In this paper, a constrained unscented Kalman filter (CUKF) algorithm is proposed to improve accuracy of numerical substructure modeling in hybrid testing. During hybrid testing, the numerical models of numerical substructures which are assumed identical to the physical substructures are updated online with the CUKF approach based on the measurement data from physical substructures. The CUKF method adopts sigma points (i.e., sample points) projecting strategy, with which the positions and weights of sigma points violating constraints are modified. The effectiveness of the proposed hybrid testing method is verified by pure numerical simulation and real-time as well as slower hybrid tests with nonlinear specimens. The results show that the new method has better accuracy compared to conventional hybrid testing with fixed numerical model and hybrid testing based on model updating with UKF.

New minimum sigma set for unscented filtering

SummaryIn this work, a new minimum set of sigma points for unscented filtering is proposed along with its unscented Kalman filter in both square‐root and nonsquare‐root forms. Comparative with the other reduced sigma sets of the literature, the new sigma set is, in some cases, better defined or, in another case, a generalization. In numerical examples, the unscented transform of the new sigma set is compared with the unscented transforms of the other reduced sigma sets of the literature. In addition, the performance of the new unscented Kalman filter is studied in an aircraft target tracking scenario. Copyright © 2014 John Wiley & Sons, Ltd.

Reduced-Rank Sigma-Point Kalman Filter and Its Application in ENSO Model

Abstract The huge computational expense has been a main challenge while applying the sigma-point unscented Kalman filter (SPUKF) to a high-dimensional system. This study focuses on this issue and presents two methods to construct a reduced-rank sigma-point unscented Kalman filter (RRSPUKF). Both techniques employ the truncated singular value decomposition (TSVD) to factorize the covariance matrix and reduce its rank through truncation. The reduced-rank square root matrix is used to select the most important sigma points that can retain the main statistical features of the original sigma points. In the first technique, TSVD is applied on the covariance matrix constructed in the data space [RRSPUKF(D)], whereas in the second technique TSVD is applied on the covariance matrix constructed in the ensemble space [RRSPUKF(E)]. The two methods are applied to a realistic El Niño–Southern Oscillation (ENSO) prediction model [Lamont-Doherty Earth Observatory model, version 5 (LDEO5)] to assimilate the sea surface temperature (SST) anomalies. The results show that both the methods are more computationally efficient than the full-rank SPUKF, in spite of losing some estimation accuracy. When the truncation reaches a trade-off between cost expense and estimation accuracy, both methods are able to analyze the phase and intensity of all major ENSO events from 1971 to 2001 with comparable estimation accuracy. Furthermore, the RRSPUKF is compared against ensemble square root filter (EnSRF), showing that the overall analysis skill of RRSPUKF and EnSRF are comparable to each other, but the former is more robust than the latter.

Robustifying optimal experiment design for nonlinear, dynamic (bio)chemical systems

Dynamic experiments that yield as much information as possible are highly valuable for estimating parameters in nonlinear dynamic processes. Techniques for model-based optimal experiment design enable to systematically design such experiments. However, these experiments depend on the current best estimate of the parameters, which are not necessarily the true values. Consequently, in real experiments (i) the information content can be lower than predicted and (ii) state constraints can be violated. This paper presents a novel, computationally tractable formulation that enables the robustification of optimally designed experiments with respect to (i) information content and (ii) constraint satisfaction. To this end, the objective function is the expected value of a scalar function of the Fisher information matrix, which is efficiently computed using the sigma point method. This approach already has a robustifying effect. The sigma point method also enables the efficient computation of constraints’ variance–covariance matrix, this can be exploited for further robustification.

Storage battery remaining useful life prognosis using improved unscented particle filter

Storage battery is one of the most important power sources in portable devices, marine systems, automotive vehicles, aerospace systems, and so on. For this kind of battery, it is essential to prognose its remaining useful life before its end of life, which would reduce some unnecessary sudden disasters caused by battery failure. In this article, we propose an improved unscented particle filter method for prognosing the remaining useful life of storage battery, in which the sigma samples of unscented transformation in traditional unscented particle filter are generated by singular value decomposition, and then, those sigma points are propagated by the standard unscented Kalman filter to generate a sophisticated proposal distribution. When both improved unscented particle filter and unscented particle filter methods were used for prognosing the remaining useful life of storage battery, it shows that the performance of improved unscented particle filter is better than unscented particle filter; the proposed method is more robust in remaining useful life prognosis procedure.

Optimal design of symmetrical/asymmetrical sigma-point Kalman filter using genetic algorithms

During the past decades, there have been a great amount of research activities in approximate problem solving to estimate the real amount of states of dynamical systems using measurement data corrupted with noise. One of the most famous approximate solving methods is Unscented Kalman Filter (UKF). In UKF, the estimation is based on a collection of symmetrical sigma points around an a priori-estimated state. In this paper, Genetic Algorithm (GA) is firstly used to optimally select the best coefficients of the symmetrical sigma points related to Scaled Unscented Transform in UKF (GA-UKF) to minimize the mean of squared error of estimations. Moreover, GA is also used to select a collection of asymmetrical sigma points (GA-ASKF) to minimize the mean of squared error of estimations together with same statistics of the mean and covariance matrix. The new idea of the asymmetric sigma points Kalman filter of this paper which are optimally found again by using GA evidently outperforms both conventional UKF and GA-UKF of this paper in estimating the states of dynamical systems.

State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter

Two attractive features of Unscented Kalman Filter (UKF) are: (1) use of deterministically chosen points (called sigma points), and (2) only a linear dependence of the number of sigma points on the number of states. However, an implicit assumption in UKF is that the prior conditional state probability density and the state and measurement noise densities are Gaussian. To avoid the restrictive Gaussianity assumption, Gaussian Sum-UKF (GS-UKF) has been proposed in literature that approximates all the underlying densities using a sum of Gaussians. However, the number of sigma points required in this approach is significantly higher than in UKF, thereby making GS-UKF computationally intensive. In this work, we propose an alternate approach, labeled Unscented Gaussian Sum Filter (UGSF), for state estimation of nonlinear dynamical systems, corrupted by Gaussian state and measurement noises. Our approach uses a Sum of Gaussians to approximate the non-Gaussian prior density. A key feature of this approximation is that it is based on the same number of sigma points as used in UKF, thereby resulting in similar computational complexity as UKF. We implement the proposed approach on two nonlinear state estimation case studies and demonstrate its utility by comparing its performance with UKF and GS-UKF.

Quadrature Kalman filter–based state of charge estimation for lithium-ion battery

This article presents an approach to lithium-ion battery state of charge estimation based on the quadrature Kalman filter. Among the existing state of charge estimation approaches, the extended Kalman filter–based state of charge and unscented filter–based state of charge algorithms are influenced by the linearization or the solution of sigma points. The proposed quadrature Kalman filter–based state of charge algorithm avoids these problems. Specifically, the battery system equations are built based on the second-order resistance–capacitance equivalent circuit model, and the parameters are identified according to the hybrid pulse power characterization discharging test. Then, the quadrature points and corresponding weights are defined by the Gauss–Hermite quadrature rule, and the Kronecker tensor product is adopted to solve the points of multivariate. In addition, the stability of quadrature Kalman filter–based state of charge is verified. Finally, the simulation is carried out under the discharging and urban dynamometer driving schedule condition, which demonstrates that the quadrature Kalman filter–based state of charge algorithm has a better performance compared with extended Kalman filter–based state of charge and unscented filter–based state of charge.

Linear dependence of the Hall coefficient of 1% Na doped PbTe with varying magnetic field

Heavily p‐doped PbTe is known to be a highly efficient thermoelectric material and theories explaining this have connected it to positive curvature in the Seebeck coefficient between 300 and 500 K and a peak in the Hall coefficient at 500 K due to band convergence. Recent work by Jaworski et al. has suggested that a pocket of high mobility electrons between the L and Sigma point in the Brillouin zone are responsible for these effects, inferred from an observed inversion of the Hall resistivity under small magnetic fields (<0.5 T). Here the Hall resistivity of 1% Na doped PbTe is measured in magnetic fields ranging from 0.1 to 2 T at 293 and 626 K and found to be linear without any inversion of Hall resistivity.

Mobile Localization in Non-Line-of-Sight Using Constrained Square-Root Unscented Kalman Filter

Localization and tracking of a mobile node (MN) in non-line-of-sight (NLOS) scenarios, based on time of arrival (TOA) measurements, is considered in this work. To this end, we develop a constrained form of square root unscented Kalman filter (SRUKF), where the sigma points of the unscented transformation are projected onto the feasible region by solving constrained optimization problems. The feasible region is the intersection of several discs formed by the NLOS measurements. We show how we can reduce the size of the optimization problem and formulate it as a convex quadratically constrained quadratic program (QCQP), which depends on the Cholesky factor of the \textit{a posteriori} error covariance matrix of SRUKF. As a result of these modifications, the proposed constrained SRUKF (CSRUKF) is more efficient and has better numerical stability compared to the constrained UKF. Through simulations, we also show that the CSRUKF achieves a smaller localization error compared to other techniques and that its performance is robust under different NLOS conditions.

Target Tracking for Robot-Fishes Based on Unscented Kalman Filter

Because of the nonlinear motion of the robot-fish and the uncertain water waves, it is difficult to track dynamic target accurately and quickly for the robot-fish. To solve this problem, this paper proposed the target tracking strategy of underwater robot-fish based on Unscented Kalman filter (UKF). UKF is a nonlinear filtering on the basis of unscented transformation, which consists of prediction and update recursively, thus estimate the state of the system. To avoid the particle degeneracy, this paper selects sigma points adopting scale symmetric sampling strategy. We apply UKF method to the underwater robot-fish competition in China. Simulations show that the Robot-fish has fast tracking speed, shorter tracking path and smaller tracking error using UKF method than not using the method. It has excellent tracking performance.

X-ray pulsar-based navigation using time-differenced measurement

In order to reduce the composite impacts of the systematic biases in the X-ray pulsar-based navigation system, an innovative navigation method is proposed. The proposed method employs the time-differenced measurement which is the difference between the measurements at the neighbor epochs. For Earth-orbiting satellite, the systematic biases vary slowly over the navigation period, and the time-differenced measurement can greatly reduce the major part of systematic biases. Through observability analysis, the corresponding navigation system is demonstrated to be completely observable. In order to solve the correlation between the process and measurement noises in the proposed method, a modified unscented Kalman filter is derived. In addition, the modified unscented Kalman filter propagates the selected sigma points to generate the time-differenced measurement model without involving linearization error. The results of simulations have shown that the proposed method can effectively reduce the composite impacts of systematic biases including the pulsar angular position error, pulsar distance error, Earth ephemeris error, and satellite-borne clock error.