Sequence Unscented Kalman Filtering Algorithm

This paper presents the efficient Unscented Kalman filter (UKF) based estimation and control of a 3-DOF Omni bundle robot. The angular trajectory tracking becomes more difficult under highly non-linear measurement noise and process noise. A rapidly converging Unscented Kalman Filter (UKF) is presented in the paper. The extended Kalman filter (EKF) provides less accurate results in the highly non-linear system and sometime may become unstable because of convergence related issues, whereas in UKF the results in the non-linear system are better than EKF. UKF is more accurate in state and noise estimation. The computed torque technique is taken as a model to apply UKF. After, estimation the noise is removed. The experiment is conducted on Omni bundle with process and measurement noise. The results prove the superiority of presented UKF over conventional UKF and EKF.

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.

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

State of Charge (SOC) estimation plays a crucial role in managing the performance and longevity of electric vehicle (EV) batteries. Various estimation techniques, including various Kalman Filters (KF), have been employed for this purpose. The mathematical process and limits of different KF family algorithms are analysed. In this study, an impact of nonlinear KF parameters on SOC estimation accuracy for EV batteries is analyzed. Specifically, the performance of Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF) is examined. EKF is most preferred for SOC estimation whereas introducing sigma points in UKF is the basic difference between EKF and UKF. In comparison to EKF and UKF, PF is more flexible since it applies to nonlinear systems with non-Gaussian distributions. Utilising a sequence of noisy measurements, KF is a recursive method that approximates the state of a dynamic system. It requires accurate parameters such as process noise covariance (Q) and measurement noise covariance (R) to work effectively. In this work, the effect of these parameters on state of charge estimation using nonlinear KFs has been analysed under different settings of noise covariance matrices. Tuning these parameters is essential for optimizing the performance of the KF and ensuring accurate and reliable state estimation in real-world applications. To find parameter values that provide the best balance between accuracy, robustness, and computational efficiency, the tuning process involves experimentation, simulation, and validation using real-world data. According to the results observed, the filter parameters such as three diagonal elements of Q are selected as 2e-8, 0.005, 0.003 and R as 2e-6. By evaluating the state estimation accuracy under different driving scenarios, robustness and sensitivity of filtering technique in estimating SOC for EV batteries has been studied. This analysis contributes to the optimization of SOC estimation algorithms, enhancing the reliability and efficiency of EV battery management systems.

Constrained nonlinear state estimation based on the UKF approach

Fast non-intrusive estimation of liquid-phase interface and boundary heat flux in participating media by the Kalman filtering methods

PurposeAutonomous orbit determination using geomagnetic measurements is an important backup technique for safe spacecraft navigation with a mere magnetometer. The geomagnetic model is used for the state estimation of orbit elements, but this model is highly nonlinear. Therefore, many efforts have been paid to developing nonlinear filters based on extended Kalman filter (EKF) and unscented Kalman filter (UKF). This paper aims to analyze whether to use UKF or EKF in solving the geomagnetic orbit determination problem and try to give a general conclusion.Design/methodology/approachThis paper revisits the problem and from both the theoretical and engineering results, the authors show that the EKF and UKF show identical estimation performances in the presence of nonlinearity in the geomagnetic model.FindingsWhile EKF consumes less computational time, the UKF is computationally inefficient but owns better accuracy for most nonlinear models. It is also noted that some other navigation techniques are also very similar with the geomagnetic orbit determination.Practical implicationsThe intrinsic reason of such equivalence is because of the orthogonality of the spherical harmonics which has not been discovered in previous studies. Thus, the applicability of the presented findings are not limited only to the major problem in this paper but can be extended to all those schemes with spherical harmonic models.Originality/valueThe results of this paper provide a fact that there is no need to choose UKF as a preferred candidate in orbit determination. As UKF achieves almost the same accuracy as that of EKF, its loss in computational efficiency will be a significant obstacle in real-time implementation.

The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.

This is the second of two consecutive papers focusing on the filtering algorithm for a nonlinear stochastic discrete-time system with linear system state equation. The first paper established a derivative unscented Kalman filter (DUKF) to eliminate the redundant computational load of the unscented Kalman filter (UKF) due to the use of unscented transformation (UT) in the prediction process. The present paper studies the error behavior of the DUKF using the boundedness property of stochastic processes. It is proved that the estimation error of the DUKF remains bounded if the system satisfies certain conditions. Furthermore, it is shown that the design of the measurement noise covariance matrix plays an important role in improvement of the algorithm stability. The DUKF can be significantly stabilized by adding small quantities to the measurement noise covariance matrix in the presence of large initial error. Simulation results demonstrate the effectiveness of the proposed technique.

Requirements of highly precise pointing performance have been imposed on recently developed spacecrafts for a variety of missions. The stringent requirements have called on on-orbit estimation of spacecraft dynamics parameters and calibration of on-board sensors as indispensible practices. Consequently, on-orbit estimation of the mass moment of inertia of spacecraft has been a major issue mostly due to the changes by solar panel deployment and a large portion of fuel consumption (Creamer et al., 1996; Ahmed et al., 1998; Bordany et al., 2000; VanDyke et al., 2004; Myung et al., 2007; Myung & Bang, 2008; Sekhavat et al., 2009). As for measurement sensors, on-board calibration of alignment and bias errors of attitude and rate sensors is one of the main concerns of attitude sensor calibration researches (Pittelkau, 2001 & 2002, Lai et al., 2003). Pittelkau (2002) proposed an attitude estimator based on the Kalman filter (Kalman, 1960), in which spacecraft attitude quaternion, rate sensor misalignment and bias, and star tracker misalignments are taken into consideration as states, whereas the body rate is dealt as a synthesized signal by the estimates. Lai at al. (2003) derived a method for alignment estimation of attitude and rate sensors based on the unscented Kalman filter (UKF) (Julier and Uhlmann, 1997). Ma and Jiang (2005) presented spacecraft attitude estimation and calibration based only magnetometer measurements using an UKF. An interesting point is that we need predesigned 3-axis excitation manoeuvres of spacecraft for both dynamics parameter estimation and sensor calibration. Therefore, this study is motivated to merge above estimation and calibration processes into a single filtering problem. It is noteworthy that poor information of moments of inertia is to be treated as a system uncertainty while the rate sensor model errors are to be incorporated into the measurement process. As a filtering algorithm, this study employs a UKF. Extended Kalman filters (EKFs) have been successfully applied to the nonlinear attitude estimation problem (Crassidis et al., 2007). Hybrid estimation using the EKF has been reported by Myung at al. (2007). However, the EKF estimates using the first order linearization, which may lead to instability of the filter (ValDyke et al., 2004). The UKF approximates the nonlinear model to the second order by spreading points 1 sigma apart from the a priori mean. Performing nonlinear

Traffic state estimation problems must deal with imperfect information. Imperfect information may occur as a result of misalignments between reality and the assumptions or simplifications made when developing the models as well as incomplete or corrupted data. One of the well-known methods of handling imperfect information is the Kalman filter algorithm and its variations, such as the extended Kalman filter and the unscented Kalman filter. While the standard Kalman filter and the extended Kalman filter have been widely applied to the state estimation of linear systems, the unscented Kalman filter has been reported the better choice in the multiclass traffic state estimation. In many applications of Kalman filters to traffic estimation problems, the model and measurement noise covariance matrices are normally estimated. When there is a mismatch between the true and the assumed noise distribution, however, the filter often suffers from performance degradation and even divergence in certain situations. To this end, this paper presents a more efficient and accurate algorithm embedded in the unscented Kalman filter to simultaneously estimate the traffic state and the model noise distribution statistics. The proposed method is facilitated through a simultaneous update of the model noise covariance matrix in the predicted covariance equations of the unscented Kalman filter algorithm. It is found through simulation that the proposed algorithm may improve the model performance over the standard algorithm.

We demonstrate experimentally that inference in a complex hybrid dynamic Bayesian network (DBN) is possible using the 2-time slice DBN (2T-DBN) from (D. Koller et al., Sequential Monte Carlo Methods in Practice: p.445-464, Springer-Verlag, NY, 2000) to model fault detection in a watertank system. In this, a generic particle filter (PF) is used for inference. We extend the experiment and perform approximate inference using The extended Kalman filter (EKF) and the unscented Kalman filter (UKF). Furthermore, we combine these techniques in a 'non-strict' Rao-Blackwellisation framework and apply it to the watertank system. We show that UKF and UKF in a PF framework outperform the generic PF, EKF and EKF in a PF framework with respect to accuracy and robustness in terms of estimation RMSE (root-mean-square error). Especially, we demonstrate the superiority of UKF in a PF framework when our beliefs of how data was generated are wrong. We also show that the choice of network structure is very important for the performance of the generic PF and the EKF algorithms, but not for the UKF algorithms. Furthermore, we investigate the influence of data noise in the watertank simulation. Theory and implementation is based on the theory presented in (R. v.d. Merwe et al., Tech. Rep. CUED/F-INFENG/TR-380, Dept. of Eng., Cambridge Uni., 2000).

In this paper, we present a generalized iterated Kalman filter (GIKF) algorithm for state estimation of a nonlinear stochastic discrete-time system with state-dependent multiplicative observation noise. The GIKF algorithm adopts the Newton-Raphson iterative optimization steps to yield an approximate maximum a posteriori estimate of the states. The mean-square estimation error (MSE) and the Cramér-Rao lower bound (CRLB) of the state estimates are also derived. In particular, the local convergence of MSE of GIKF is rigorously established. It is also proved that the GIKF yields a smaller MSE than those of the generalized extended Kalman filter and the traditional extended Kalman filter. The performance advantages and convergence of GIKF are demonstrated using Monte Carlo simulations on a target tracking application in a range measuring sensor network.

The extended Kalman filter (EKF) has been used widely in Global Position System (GPS) and Strapdown Inertial Navigation Systems (SINS) integrated navigation systems,which simply linearizes all nonlinear models.However, careful treatment of the nonlinearity of the system models is particularly critical when the integrated systems use low cost micro-electro-mechanical (MEMS) inertial sensors. Hence application of the EKF in MEMS inertial sensor-based integrated navigation systems will likely lead to inaccurate results. To overcome the shortcomings of the EKF, the unscented Kalman filter (UKF) has been proposed.The UKF does not require the linearization of the system models. Alternatively it uses a set of deterministically selected "sigma-points",which completely capture the true mean and covariance of the original random vector. Then these sigma-points are propagated through the nonlinear models.This captures the mean and covariance to second order accuracy for arbitrary nonlinear functions. In this paper the EKF and UKF are applied in GPS/SINS integrated systems respectively. Simulation result demonstrates that the performance of the UKF is better than the EKF in MEMS inertial sensor-based SINS/GPS integrated systems.

Effect of Noises on the Performance of Nonlinear State Estimator for Semi Batch Autocatalytic Esterification Reactor