Tracking with nonlinear measurement model by coordinate rotation transformation

The problem of tracking with very long range radars is studied in this paper. First, the measurement conversion from a radar's r-u-v coordinate system to the Cartesian coordinate system is discussed. Although the nonlinearity of this coordinate transformation appears insignificant based on the evaluation of the bias of the converted measurements, it is shown that this nonlinearity can cause significant covariance inconsistency in the conventionally converted measurements (CM1). Since data association depends critically on filter consistency, this issue is very important. Following this, it is shown that a suitably corrected conversion (CM2) eliminates the inconsistency. Then, initialized with the converted measurements (using CM2), four Cartesian filters are evaluated. It is shown that, among these filters, the converted measurement Kalman filter with second order Taylor expansion (CM2KF) is the only one that is consistent for very long range tracking scenarios. Another two approaches, the range-direction-cosine extended Kalman filter (ruvEKF) and the unscented Kalman filter (UKF) are also evaluated and shown to suffer from consistency problems. However, the CM2KF has the disadvantage of reduced accuracy in the range direction. To fix this problem, a consistency-based modification for the standard extended Kalman filter (E1KF) is proposed. This leads to a new filtering approach, designated as measurement covariance adaptive extended Kalman filter (MCAEKF). For very long range tracking scenarios, the MCAEKF is shown to produce consistent filtering results and be able to avoid the loss of accuracy in the range direction. It is also shown that the MCAEKF meets the posterior Carmer-Rao lower bound for the scenarios considered.

In radar tracking the target motion is best modeled in Cartesian coordinates. Its position is however measured in polar coordinates (range and azimuth). Tracking in Cartesian coordinates with noisy polar measurements requires either converting the measurements to a Cartesian frame of reference and then applying the linear Kalman filter to the converted measurement or using the extended Kalman filter (EKF) in mixed coordinates. The first approach is accurate only for moderate cross-range errors; the second approach is consistent only for small errors. A new efficient tracking algorithm using the multidimensional Gauss-Hermite quadratures to propagate the mean and the covariance of the conditional probability density function is presented. This method is compared with the EKF and the converted measurement Kalman filter (CMKF) and it is shown to be more accurate.

In a multi-sensor system, sensors often provide data at different rates with different communication delays. To the asynchronous fusion for target tracking with infrared and laser sensors, that provide range and bearing information respectively, there mainly exist two problems need to be solved. One is temporal registration of different sensors since infrared sensor has a shorter sample time than laser sensor. The measurements are dealt with least square method, which makes full use of the high frequency characteristic of infrared sensor. The other is the choice of filtering algorithm because of the correlation of coordinate transformation when measurement equation is described in Cartesian coordinate. Converted measurement Kalman filter (CMKF) is a popular solution to the nonlinear estimation problem expect nonlinear filtering algorithm. Simulations of the algorithms adopted in the paper were carried out to track the target and the performance of different algorithms was compared. The results show that the algorithm based on CMKF yields better track performance than the fusion algorithm based on standard Kalman filter (KF) and a little worse than the algorithm based on unscented Kalman filter (UKF).

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

We present in this work the use of the extended Kalman filter (EKF) and unscented Kalman filter (UKF) for identification of constitutive material parameters with application in mechanized tunneling. Although both filters are based on the principle of recursive least squares estimation, one differs from another in terms of where approximation is made. Whereas in the EKF first‐order Taylor series expansion is used to approximate the nonlinear modeling equation, in the UKF approximation of the probability density of the state is made using a small number of well defined points. To validate the methods, we performed parameter identification of the Hardening Soil constitutive model used for describing the soil behavior in an tunnel excavation model. Both methods showed fast and stable convergence of the considered soil parameters ‐ the four parameters of the Hardening Soil model. Although the EKF requires less number of forward calculations of the numerical model, the UKF is favored since it does not require calculation of the derivatives of the observables with respect to the identifying parameters. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

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.

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.

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.

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.

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.

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

Flying vehicle’s navigation, direction, and control in real‐time results in the design of a strap‐down inertial navigation system (INS). The strategy results in low accuracy, performance with correctness. Aiming at the attitude estimation problem, many data fusion or filtering methods had been applied, which fail in many cases, which attains the nonlinear measurement model, process dynamics, and high navigation range. The main problem in unmanned aerial vehicles (UAVs) and flying vehicles is the determination of attitude angles. A novel attitude estimation algorithm is proposed in this study for the unmanned aerial vehicle (UAV). This research article designs two filtering algorithms for fixed‐wing UAVs which are nonlinear for the attitude estimation. The filters are based on Kalman filters. The unscented Kalman filter (UKF) and cubature Kalman filter (CKF) were designed with different parameterizations of attitude, i.e., Euler angle (EA) and INS/unit quaternion (UQ) simultaneously. These filters, EA‐UKF and INS‐CKF, use the nonlinear process and measurement model. The computational results show that among both filters, the CKF attains a high accuracy, robustness, and estimation for the attitude estimation of the fixed‐wing UAV.

The stochastic nature of noise signals affects the vehicle’s internal states and the outputs, resulting in the difficulty in estimation. The unknown or time-varying nature of noise signals if not taken into account for estimation, the results will diverge and be highly deteriorated. In this paper, a maximum likelihood principle (MLP) based adaptive robust extended kalman filter for estimating the states of the adopted non-linear vehicle model is proposed. An observability test is done for the purpose of estimation. A covariance matching (CM) based robust adaptive high forgetting factor based fault tolerant technique is also employed on the robust adaptive extended and unscented kalman filters for comparison purpose. The Robustness of the filter is analyzed by varying the parameter of the vehicle through a local sensitivity analysis. The results show that the MLP based approach to the extended kalman filter performs well in three simulations for sinusoidal steering, Double Lance Change, J-Turn, Fishhook, Slalom maneuver in comparison to robust adaptive unscented kalman filter. Friction coefficient of 0.8 (dry road) and 0.4 (wet road) is chosen for the simulation. The sideslip angle RMSE value for MLP based estimation is obtained as 2.62e-05, 4.545e-06 for Sine and DLC maneuver.

For land-based navigation, Euler angles are often used in INS/GPS integrated navigation systems. However, the trigonometric operations required in the updates and forming of the rotation matrices for transforming the INS measurements from the body frame to the navigation frame turns the system model to be highly nonlinear. Besides, using low-cost MEMS-based IMUs, the gyroscope bias errors must be correctly estimated and compensated, which makes the nonlinearity problem a critical one. In this contribution, three Kalman filtering methods (i.e., Extended Kalman filter with simplified system model, Extended Kalman filter with linearized system model and Unscented Kalman filter with nonlinear system model) are utilized in INS/GPS tightly-coupled integration. Simulations and field experiments are conducted. Numerical results are compared in terms of both estimation accuracy and processing time.