Data Fusion Issues Associated with Multiple-Radar Tracking Systems

SummaryThis paper considers the state estimation problem of bilinear systems in the presence of disturbances. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. It is well known that the extended Kalman filter (EKF) is proposed based on the Taylor expansion to linearize the nonlinear model. In this paper, we show that the EKF method is not suitable for bilinear systems because the linearization method for bilinear systems cannot describe the behavior of the considered system. Therefore, this paper proposes a state filtering method for the single‐input–single‐output bilinear systems by minimizing the covariance matrix of the state estimation errors. Moreover, the state estimation algorithm is extended to multiple‐input–multiple‐output bilinear systems. The performance analysis indicates that the state estimates can track the true states. Finally, the numerical examples illustrate the specific performance of the proposed method.

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.

The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.

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.

Data fusion algorithms are inevitable in improving the accuracy and reliability of radar target tracking. Considering the shortcomings of target tracking in Cartesian coordinates, all the fusion process are carried out in spherical coordinate system. To fuse data from bidimensional radar and Infrared Search and Track (IRST), five data fusion methods, Bayesian fusion, Signal variance based fusion, Optimal fusion, Error variance based fusion and Extended Kalman Filter (EKF) based fusion are proposed and analyzed. The performances of the methods are analyzed in terms of Root Mean Square Error in Position (RMSPE) and Mean Absolute Error in Position (MAPE) for a wide range of measurement noise variances. It is concluded that EKF based fusion has the best error performance. Kalman Filter State-Vector Fusion (KF-SVF) which is a hierarchical fusion architecture, having first level of fusion with Kalman filter and second level with error variance based fusion method, is available in the literature. In this work it has been expanded to Extended Kalman Filter—State-Vector Fusion (EKF-SVF) with either error variance based fusion, signal variance based fusion or optimal information fusion as second level. All the three EKF-SVF methods have similar error characteristics, but EKF-SVF with signal variance based fusion, unlike the other two, provides less computational cost since it does not involve matrix inversion. While comparing EKF based fusion and EKF-SVF, EKF-SVF offers increased error minimization.

Wireless sensor networks can be used for the localization and tracking of moving targets. However, the range measurements are noisy and Kalman filters are frequently used to improve the tracking accuracy. Three different tracking algorithms, namely, a standard Kalman filter (SKF), an extended Kalman filter (EKF), and a modified Kalman filter (MKF) are empirically studied in terms of accuracy and latency for a range-based indoor tracking system. The experimental results show that the filtering techniques improve the tracking accuracy when the target is moving rapidly. However, different forms of Kalman filters introduce different levels of latency which affects the real-time tracking performance.

The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy. The object of research is a mathematical model describing the movement of a robot with five degrees of freedom for a warehouse. The work was aimed at analyzing the kinematic structure of the manipulator, on the basis of which the base and local coordinate systems were selected, as well as further formalized recording of the kinematic equations in matrix form. It is noted that one of the most problematic places is that the algorithms for controlling the robot most often contain local rules for the interaction of robots between themselves and the external environment, and emergent behavior is manifested as a result of the application of these rules, which does not have a formal description. Therefore, it is important to modernize the models describing the motion of a robot with five degrees of freedom for a warehouse. Using the matrix method, the sequence of constructing coordinate systems is described and its mathematical description is given, which will make it possible to eliminate this drawback in the future. The computer implementation of the developed algorithms was carried out using methods for processing matrix data structures. The principle of constructing a kinematic model of a robot is presented, with the help of which the main coordinate transformation matrices are obtained for robot with five degrees of freedom, and the possibility of taking into account the size of the gaps in the joints is shown. The resulting model is obtained, which is proposed for use in building control algorithms for a robot with an automatic gap selection, as well as in robot calibration. This is due to the fact that the proposed model has a number of features, in particular, the basic coordinate system and the coordinate system of each link of robot with five degrees of freedom are taken into account. This makes it possible to obtain the values of the indicators for the projection of the robot position vector in the initial state, in the rotation of the fourth link at a well-defined angle and in the case of a vertically straightened manipulator. Compared to similar known studies, this provides advantages such as minimizing errors in position, speed and motion accuracy.

Spatial arbitrarily curved microbeams with the modified couple stress theory: Formulation of equations of motion

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.

Coherent propagation of polarized millimeter waves through falling nonspherical hydrometeors is considered on the basis of Foldy's approximation. It is assumed that the distribution of hydrometeors over orientations is locally axially symmetric, the axis of symmetry being given by the local direction of air flow around the hydrometeors. An efficient rigorous method is described to compute the orientationally averaged forward-scattering amplitude matrix. This method is based on Waterman's T-matrix approach and fully exploits rotational properties of the T-matrix. First, the T-matrix is analytically averaged over hydrometeor orientations in the local coordinate system with the Z -axis along the direction of local air flow. Then, the elements of the forward-scattering amplitude matrix with respect to the local and laboratory coordinate systems are calculated via simple analytical expressions. Analytical solutions of the propagation equation for the coherent electric field are discussed. Cross polarization of...

This letter proposes a shooting and bouncing ray (SBR) method based on a dynamic octree structure to accelerate the solution of scattering from a moving target. In this algorithm, three coordinate systems are established, i.e., the world coordinate system (WCS), the local coordinate system (LCS), and the perspective projection coordinate system (PPCS). First, an octree structure is created to store the target facets in the LCS that moves with the target in the WCS. The octree remains relatively stationary with the target, which avoids rebuilding the structure at every moment. Second, to accelerate the shadow elimination, the scanline algorithm is used to convert the 3-D processing into a 2-D graphic display in the PPCS. Third, high-order ray tracing is completed in the LCS. A forward–backward ray-tracing technology is adopted to guarantee the precision of the illuminated facet identification. Finally, the information of illuminated facets is transferred from the LCS to the WCS, and the scattering of target is calculated in the WCS. Meanwhile, some numerical simulations of our method are compared with those of the FEKO-MLFMM and the traditional octree-SBR method. The results show that the proposed method greatly reduces the computation time while maintaining satisfactory accuracy.

This study presents an adaptation of a novel estimation methodology to the general nonlinear three-dimensional problem of tracking a maneuvering target. The two-step optimal estimator (TSE) suggests an attractive alternative to the standard extended Kalman filter (EKF). A superior performance is accomplished by dividing the estimation problem into two steps: a linear first step and a nonlinear second step. The target tracking performance of the TSE is shown to be better than an EKF implemented in either inertial or modified spherical coordinates. In the passive case, where bearing/elevation angles only are measured, the TSE yields excellent range and target acceleration estimates. In the active case, where range measurement is available as well, a homing missile employing closed-loop optimal guidance based on the TSE state estimates obtains smaller miss distances than with either versions of the EKF.

In multipath assisted positioning, each multipath component arriving at a receiver is regarded as being sent by a virtual transmitter in a line-of-sight condition. These virtual transmitters can be used for localizing a user. In the general case, their locations are unknown. With simultaneous localization and mapping (SLAM), the locations of physical and virtual transmitters can be estimated jointly with the location of a user. However, the convergence time can be high. In scenarios such as malls or museums, multiple users travel through the same scenario on different trajectories. Different users can cooperate by exchanging maps of physical and virtual transmitters in order to decrease the convergence time and increase the positioning performance. Though, each user is in its own local coordinate system. When a map of transmitters is handed from one user to another, the relative rotation and translation of the coordinate systems of the two users need to be estimated based on the transmitters in the two users’ maps. Within this paper, we propose an algorithm to merge and update an obtained transmitter map with the transmitters estimated by the current user. We evaluate our proposed method by simulations in an indoor mall. In a crowdsourcing based approach, maps are handed from one user to another, updated, and handed to the next user. Our evaluations show that exchanging transmitter maps improves the positioning performance, although each user is in its own local coordinate system not knowing the parameters relating its own coordinate system to the ones of obtained transmitter maps.

Nonlinear state estimation problem is an important and complex topic, especially for real-time applications with a highly nonlinear environment. This scenario concerns most aerospace applications, including satellite trajectories, whose high standards demand methods with matching performances. A very well-known framework to deal with state estimation is the Kalman Filters algorithms, whose success in engineering applications is mostly due to the Extended Kalman Filter (EKF). Despite its popularity, the EKF presents several limitations, such as exhibiting poor convergence, erratic behaviors or even inadequate linearization when applied to highly nonlinear systems. To address those limitations, this paper suggests an improved Extended Kalman Filter (iEKF), where a new Jacobian matrix expansion point is recommended and a Frobenius norm of the cross-covariance matrix is suggested as a correction factor for the a priori estimates. The core idea is to maintain the EKF structure and simplicity but improve its accuracy. In this paper, two case studies are presented to endorse the proposed iEKF. In both case studies, the classic EKF and iEKF are implemented, and the obtained results are compared to show the performance improvement of the state estimation by the iEKF.

This paper presents a variable-stepsize unscented Kalman filter for treating continuous-time stochastic models in radar tracking, numerically. Our method is grounded in the Gauss-type nested implicit Runge-Kutta formula of order 6 applied for solving moment differential equations (MDEs). The built-in global error control ensures that the MDEs are integrated with negligible errors. The latter raises the accuracy of state estimation and makes our state estimator competitive to the accurate continuous-discrete extended Kalman filter (ACD-EKF) and the mixed-type accurate continuous-discrete extended-unscented Kalman filter (ACD-EUKF) developed earlier for target tracking models. The effectiveness of the new filter and its comparison to the ACD-EKF and ACD-EUKF are studied on a 7-dimensional radar tracking problem with both short and long waiting times. Our research suggests that the unscented Kalman filtering is more robust to the error accumulation and better suits for treating target tracking models with long sampling periods whereas the extended Kalman filtering is more accurate when the waiting time is short.