Efficient Approximation of Kalman Filter for Target Tracking

A Kalman filter in the Cartesian coordinates is described for a maneuvering target when the radar sensor measures range, bearing, and elevation angles in the polar coordinates at high data rates. An approximate gain computation algorithm is developed to determine the filter gains for on-line microprocessor implementation. In this approach, gains are computed for three uncoupled filters and multiplied by a Jacobian transformation determined from the measured target position and orientation. The algorithm is compared with the extended Kalman filter for a typical target trajectory in a naval gun fire control system. The filter gains and the tracking errors for the proposed algorithm are nearly identical to the extended Kalman filter, while the computation requirements are reduced by a factor of four.     

Best linear unbiased filtering with nonlinear measurements for target tracking

In tracking applications, target dynamics are usually modeled in the Cartesian coordinates, while target measurements are directly available in the original sensor coordinates. Measurement conversion is widely used such that the Kalman filter can be applied in the Cartesian coordinates. A number of improved measurement-conversion techniques have been proposed recently. However, they have fundamental limitations, resulting in performance degradation, as pointed out in a recent survey conducted by the authors. A filter is proposed here that is theoretically optimal in the sense of minimizing the mean-square error among all linear unbiased filters in the Cartesian coordinates. The proposed filter is free of the fundamental limitations of the measurement-conversion approach. Results of an approximate, recursive implementation are compared with those obtained by two state-of-the-art conversion techniques. Simulation results are provided.     

Coordinate Conversion and Tracking for Very Long Range Radars

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.     

A Tracking Filter in Spherical Coordinates Enhanced by De-noising of Converted Doppler Measurements

Tracking problem in spherical coordinates with range rate (Doppler) measurements, which would have errors correlated to the range measurement errors, is investigated in this paper. The converted Doppler measurements, constructed by the product of the Doppler measurements and range measurements, are used to replace the original Doppler measurements. A de-noising method based on an unbiased Kalman filter (KF) is proposed to reduce the converted Doppler measurement errors before updating the target states for the constant velocity (CV) model. The states from the de-noising filter are then combined with the Cartesian states from the converted measurement Kalman filter (CMKF) to produce final state estimates. The nonlinearity of the de-noising filter states are handled by expanding them around the Cartesian states from the CMKF in a Taylor series up to the second order term. In the mean time, the correlation between the two filters caused by the common range measurements is handled by a minimum mean squared error (MMSE) estimation-based method. These result in a new tracking filter, CMDN-EKF2. Monte Carlo simulations demonstrate that the proposed tracking filter can provide efficient and robust performance with a modest computational cost.     

Scan conversion of radar images

The object of scan conversion is to resample a polar-sampled image into a Cartesian format while retaining integrity of the geometry and content of the image. Linear interpolation and decimation in two dimensions in the context of scan conversion are reviewed. Appropriate approximations of the nonlinear relationship between Cartesian and polar coordinates are then made to allow the application of the linear interpolation theory to produce a number of practical scan conversion algorithms. Together with an image prefiltering process, which is applied to areas which require scan conversion decimation, an effective scheme is presented     

False alarm effects on estimation in multitarget trackers

An analysis of false alarm effects on tracking filter performance in multitarget track-while-scan radars, using variable correlation gates, is presented. The false alarms considered originate from noise, clutter, and crossing targets. The dimensions of the correlation gates are determined by filter prediction and measurement error variances. Track association is implanted either by means of a distance weighted average of the observations or by the nearest neighbor rule. State estimation is performed by means of a second-order discrete Kalman filter, taking into consideration random target maneuvers. Measurements are made in polar coordinates, while target dynamics are estimated in Cartesian coordinates, resulting in coupled linear filter equations. the effect of false alarms on the observation noise covariance matrix, and hence on state estimation errors, is analyzed. A computer simulation example, implementing radar target tracking with a variable correlation gate in the presence of false alarms, is discussed     

Alpha-beta filters in polar coordinates with accelerationcorrections

The use of polar coordinates is sometimes computationally advantageous for tracking, but complications arise because the position of constant velocity targets is no longer a linear function of time as it is for cartesian coordinates. However, this difficulty can be avoided by using pseudoacceleration correction factors which are added to the prediction equations to give approximately correct system dynamics, but at the expense of an increase in system noise. For alpha-beta tracking filters, these correction factors can be included with minimal degradation in the steady-state error performance of the filter while simultaneously providing substantial reductions in bias errors     

Multiradar Tracking System Using Radial Velocity Measurements

Multiradar tracking using both position and radial velocity measurements is discussed. The measurement of two or more different radial velocity components allows the calculation of rectangular velocity components. The measurement noise of the velocity components is filtered using a Kalman filter in the same way as the Cartesian position components. Before the conversion of velocity components from radial to Cartesian coordinates, the radial velocities are aligned on a time scale to account for the time shift of the radar measurements. In order to compare multiradar tracking system performance with and without radial velocity, some simulation tests have been performed for typical paths. The simulation results show a significant improvement when radial velocity is used for tracking.     

基于天基光学监视的空间目标MSC-SPKF跟踪方法

在对天基光学监视跟踪特点进行分析的基础上,提出样点状态转换修正球坐标Sigma点卡尔曼滤波(MSC-SPKF)方法以实现对空间目标的三维被动跟踪。利用修正球坐标系和无迹变换(UT)分别在仅测角被动跟踪和获取非线性系统状态二阶矩方面的优势,选择在修正球坐标状态空间内进行Sigma点采样,并利用状态空间转换完成样点矢量非线性预测,最后在修正球坐标空间内完成滤波更新,并给出相应的初始化方法。仿真结果表明该方法在收敛性和稳健性上更有优势。     

Linear Regression Filtering and Prediction for Tracking Maneuvering Aircraft Targets

A Cartesian coordinate linear regression filter is utilized for tracking maneuvering aircraft targets. Measurements of target position are made in a line-of-sight coordinate frame, but filtering is performed in Cartesian coordinates. Numerical results are given for optimizing the truncation time constant such that a good balance is obtained between the dynamic errors and the standard deviations. Lower bounds on the dynamic errors are established for the Cartesian coordinate linear regression filter and compared with a line-of-sight coordinate Kalman filter.     

有限差分法的坐标变换诱导误差

有限差分法应用于具有复杂外形的网格时需要进行坐标变换,在此过程中经常会引入坐标变换诱导误差。在柱坐标系下使用均匀网格进行均匀流场计算,计算结果表明,即使物理坐标对计算坐标的变换函数连续可导、计算过程中坐标变换系数直接采用准确的解析式、网格完全正交并且充分光滑,也无法避免坐标变换诱导误差。理论分析表明,产生坐标变换诱导误差的机理是笛卡尔坐标系下的守恒型欧拉方程变换至贴体坐标系下后增加了源项。针对该问题,目前国内外学者通常采用几何守恒律,构建与差分格式相匹配的坐标变换系数计算方法来消除源项。本文介绍了从包含源项的离散等价方程基础上直接进行离散的新算法,在此基础上针对非等距网格条件下MUSCL类格式重构过程进行误差分析,理论推导表明重构中需要考虑非等距插值公式的影响系数,将变量转换至计算空间内进行MUSCL重构才能保证该过程具有均匀网格下的插值精度。通过理论分析及数值实验证明新算法对于均匀流场完全不会引入坐标变换误差。     

拟合法轨迹控制及其实现

本文阐述了拟合法轨迹控制的基本原理、方法及应用。提出了在直角坐标系中用多段直线拟合曲线的方法,在极坐标系中用多段阿基米德螺线拟合曲线的方法。并将此方法应用到三自由度球坐标机械手上。实验证明,拟合法轨迹控制的方法是正确可行的。     

A pure-Cartesian formulation for tracking filters

A pure-Cartesian formulation is presented for angle-only and angle-plus-range tracking filters. Unlike conventional angle-only filters, which use target elevation and bearing as measurements, the filter expresses the sensor measurements in Cartesian coordinates. Consequently, the filter performs equally well for any line-of-sight (LOS) geometry, even when target elevation approaches or is equal to ±90°     

Approximate analytical solutions of stability problems for skew plates under combined loading

The known analytical solutions of the linearized stability problems for rectangular hinged plates under the combined loading are generalized to the similar problems for skew plates, the deformation mechanics of which is described by the equations of the classical theory formed in the oblique Cartesian coordinates.     

Multiframe weak target track-before-detect based on pseudo-spectrum in mixed coordinates

Traditional multiframe Track-Before-Detect(TBD) may incur adverse integration loss resulting from model mismatch in sensor coordinates. Its suboptimal integration strategy may cause target envelope degradation. To address these issues, a pseudo-spectrum-based multiframe TBD in mixed coordinates is proposed firstly. The data search for energy integration is conducted based on an accurate model in the x-y plane while target energy is integrated based on pseudo-spectrum in sensor coordinates. The a...     

Pattern control for large-scale spacecraft swarms in elliptic orbits via density fields

Space swarms, enabled by the miniaturization of spacecraft, have the potential capability to lower costs, increase efficiencies, and broaden the horizons of space missions. The formation control problem of large-scale spacecraft swarms flying around an elliptic orbit is considered. The objective is to drive the entire formation to produce a specified spatial pattern. The relative motion between agents becomes complicated as the number of agents increases. Hence, a density-based method is adopted...     

SEQUENTIAL ALGORITHM FOR MULTISENSOR PROBABILISTIC DATA ASSOCIATION

Jointprobabillsticdataassociation(JPDA)isanalgorithmusedinsinglesensormultipletargettrackingsystems.Itemploysthenon-uniqueassignmentof"allneighbor"strategytoadaptforthedensemultitargettrackingenvironments[1].Becauseofitswideapplications,itisnecessarytoextendJPDAintosomemultiplesensortrackingsystems.Suchamultisensorsystem,forexample,canbeformedbycollocatingradarandinfraredsearchandtrack(IRST)whichcantakeadvantagesofboththesensorsbodatafusion.Undertheconditionofthesamesensors,acommonmeasure…     

极坐标算法对飞机CCIP弹道解算的优化

采用极坐标算法对某型飞机CCIP攻击模式进行弹道解算优化,该算法进行坐标变换和积分变量转换后,使弹道的积分变为一个具有确定积分上限的定积分,有效地简化了计算过程,节省了计算时间,提高了计算精度,能较好满足火控系统的精度和实时性要求。     

Automatic detection and tracking of a small surface watercraft in shallow water using a high-frequency active sonar

A high-frequency (HF) active sonar can be used to detect and track a small fast surface watercraft in shallow water based on the evolution of the watercraft wake observed in the sonar image sequence. An automatic detection and tracking (ADT) algorithm is described for this novel application. For each ping, the measurement of the target's polar position consists of 2 steps. First, the target bearing is estimated by finding the direction of arrival of the cavitation noise emitted by the watercraft. Then range measurements are extracted from the range profile (constant-angle cut of the sonar image) at the estimated target bearing. Range normalization and clutter map processing are used to reduce the number of false measurements. Estimates of the target's Cartesian position and velocity are updated at the sonar pulse repetition rate using the Kalman filter with debiased consistent converted measurements and nearest neighbour data association. The proposed algorithm is demonstrated using real data.     

用保角转绘坐标解二维和轴对称可压流Euler方程

本文推导出用保角转绘正交曲线坐标表示的二维与轴对称Euler方程,方程中的未知量以曲线坐标方向的速度分量表示,并有不含未知量导数的源项。方程组的形式与直角坐标系中的接近,因而简化了计算。本文将该方程组用于计算二维任意翼型和旋成体轴对称绕流,采用分辨率较高的Eberle特征平均显式有限体积差分算法求解,在不同区域中用不同的显式推进步数加速收敛。本方法已分别对翼型和旋成体轴对称绕流编制了计算程序,可用于自由流为亚声速、跨声速和超声速的绕流计算。大量计算结果表明,本方法精度好,激波分辨率高,编程序简单。