Manoeuvring target tracking in clutter using particle filters

A particle filter (PF) is a recursive numerical technique which uses random sampling to approximate the optimal solution to target tracking problems involving nonlinearities and/or non-Gaussianity. A set of particle filtering methods for tracking and manoeuvering target in clutter from angle-only measurements is presented and evaluated. The aim is to compare PFs to a well-established tracking algorithm, the IMM-PDA-EKF (interacting multiple model, probabilistic data association, extended Kalman filter), and to provide an insight into which aspects of PF design are of most importance under given conditions. Monte Carlo simulations show that the use of a resampling scheme which produces particles with distinct values offers significant improvements under almost all conditions. Interestingly, under all conditions considered here,using this resampling scheme with blind particle proposals is shown to be superior, in the sense of providing improved performance for a fixed computational expense, to measurement-directed particle proposals with the same resampling scheme. This occurs even under conditions favourable to the use of measurement-directed proposals. The IMM-PDA-EKF performs poorly compared with the PFs for large clutter densities but is more effective when the measurements are precise.     

Observations of Vertical Reflections from the Topside Martian Ionosphere

The Martian ionosphere has for the first time been probed by a low frequency topside radio wave sounder experiment (MARSIS) (Gurnett et al., 2005). The density profiles in the Martian ionosphere have for the first time been observed for solar zenith angles less than 48 degrees. The sounder spectrograms typically have a single trace of echoes, which are controlled by reflections from the ionosphere in the direction of nadir. With the local density at the spacecraft derived from the sounder measurements and using the lamination technique the spectrograms are inverted to electron density profiles. The measurements yield electron density profiles from the sub-solar region to past the terminator. The maximum density varies in time with the solar rotation period, indicating control of the densities by solar ionizing radiation. Electron density increases associated with solar flares were observed. The maximum electron density varies with solar zenith angle as predicted by theory. The altitude profile of electron densities between the maximum density and about 170m altitude is well approximated by a classic Chapman layer. The neutral scale height is close to 10 to 13 km. At altitudes above 180 km the densities deviate from and are larger than inferred by the Chapman layer. At altitudes above the exobase the density decrease was approximated by an exponential function with scale heights between 24 and 65 km. The densities in the top side ionosphere above the exobase tends to be larger than the densities extrapolated from the Chapman layer fitted to the measurements at lower altitudes, implying more efficient upward diffusion above the collision dominated photo equilibrium region.     

Interlaced Kalman filtering of 3D angular motion based on Euler'snonlinear equations

A novel Kalman filtering technique is presented that reduces the mean-square-error (MSE) between three-dimensional (3D) actual angular velocity values and estimated ones by an order of magnitude (when compared with the MSE resulting from direct measurements) even under extremely low signal-to-noise ratio conditions. The filtering problem is nonlinear in nature because the dynamics of 3D angular motion are described by Euler's equations. This nonlinear set of differential equations state that the angular acceleration in one axis is proportional to the torque applied to that axis, and to the products of angular velocity components in the other two axes of rotation. Instead of using extended Kalman filtering techniques to solve this complex problem, the authors developed a new approach where the nonlinear Euler's model is decomposed into two pseudolinear models (primary and secondary). The first model describes the time progression of the state vector containing the linear terms, while the other characterizes the propagation of the state vector containing the nonlinearities. This makes it possible to run two interlaced discrete-linear Kalman filters simultaneously. One filter estimates the values of the state vector containing the linear terms, while the other estimates the values of the state vector containing the nonlinear terms in the system. These estimates are then recombined, solving the nonlinear estimation process without linearizing the system. Thus, the new approach takes advantage of the simplicity, computational efficiency and higher convergence speed of the linear Kalman filter form and it overcomes many of the drawbacks typical of conventional extended Kalman filtering techniques. The high performance and effectiveness of this method is demonstrated through a computer simulation case study     

Quadratic extended Kalman filter approach for GPS/INS integration

GPS/INS integration system has been widely applied for navigation due to their complementary characteristics. And the tightly coupled integration approach has the advantage over the loosely coupled approach by using the raw GPS measurements, but hence introduces the nonlinearity into the measurement equation of the Kalman filter. So the typical method for navigation using measurements of range or pseudorange is by linearizing the measurements in an extended Kalman filter (EKF). However, the modeling errors of the EKF will cause the bias and divergence problems especially under the situation that the low quality inertial devices are included. To solve this problem, a quadratic EKF approach by adding the second-order derivative information to retain some nonlinearities is proposed in this paper. Simulation results indicate that the nonlinear terms included in the filtering process have the great influence on the performance of integration, especially in the case that the low quality INS is used in the integrated system. Furthermore, a two-stage cascaded estimation method is used, which circumvents the difficulty of solving nonlinear equations and greatly decreases the computational complexity of the proposed approach, so the quadratic EKF approach proposed in this paper is of great value in practice.     

Robustness of Velocity Feedback Controllers for Flexible Spacecraft

The robustness properties of collocated velocity feedback controllers used for damping enhancement in large, flexible space structures are investigated. It is proved that the closed-loop system using such controllers is asymptotically stable in the large when: 1) unmodeled linear time-invariant dynamics (such as sensors/actuators) are present, provided that the phase angle of such dynamics is between -90° and 90°, 2) time-varying or invariant nonlinearities lying in the first and the third quadrant (i. e., belonging to the (0, ?) sector) are present, or 3) sector-type nonlinearities following first-order stable dynamics are present in the feedback loop.     

Multi-h CPM遥测信号的定时频率联合估计技术

针对Multi-h CPM（Multi-h Continue Phase Modulation,多指数连续相位调制）遥测接收机的符号定时和载波频率同步问题,在分析MLSD（Maximum-likelihood Sequence Detection,最大似然序列检测）度量值是符号定时偏差和频率偏差的凸函数的基础上,提出一种基于迟/早门和升/降频门的符号定时和频率同步联合估计算法,通过似然值比较,对符号定时偏差和频率偏差进行迭代估计;并对该算法进行仿真,得到不同参数下的同步性能,给出同步参数选择建议.仿真结果表明：将该算法应用到Multi-h CPM遥测系统中,可实现低信噪比信号的快捕和高精度同步.     

Nonlinear Filter Evaluation for Estimating Vehicle Position and Velocity Using Satellites

Approximate nonlinear filtering theory is applied to the estimation of vehicle position and velocity in three demensions using sequential range measurements to three known locations. The particular case studied is a satellite air traffic control system which utilizes range measurements to two geostationary satellites and an altitude measurement. Three approximate filters are examined as suboptimal realizations of the minimum-variance filter and simulation results are presented to show that simple first-order approximation is an adequate representation. The parametric relationship between state covariance, measurement noise, vehicle maneuver structure, data rate, and system geometry is presented.     

Nonlinear Estimation with Radar Observations

The nonlinearities of the radar measurement model equation are examined and their influence upon the accuracy of filtering and smoothing is determined. A simple algorithm by which the effects from these nonlinearities can be significantly reduced is derived. It consists of processing the radar observations in this order: azimuth, elevation, and range, using the azimuth residual to evaluate the elevation residual, and then using this combined result to evaluate the range residual. The accuracy of estimates obtained via the algorithm is compared with and shown to be superior to that of the extended Kalman filter (EKF). Furthermore, use of the algorithm does not increase the computational complexity of estimation beyond that of the EKF.     

Measurements of L-band inland-water surface-clutter Doppler spectra

Although radar surface-clutter reflectivities from terrain are generally much greater than those from water, strong Bragg resonances at low but non-zero Doppler frequencies in backscatter from small inland bodies of water might potentially cause false alarms for moving target indicator (MTI) or other Doppler signal-processing techniques designed for target detection in ground clutter. To provide data for investigating this concern, measurements of L-band radar backscatter were recorded from the surface of a small inland freshwater reservoir in central Massachusetts. These measurements were of unusually high system stability and spectral purity so as to provide up to 80 dB of available spectral dynamic range. Strong Bragg spikes occurred in the clutter Doppler spectra from the reservoir at low (3 to 4 Hz) but non-zero Doppler frequencies. This strong Bragg resonance was persistent in time and space throughout the measurements. Spectral results are presented for all four combinations of linear polarization. Comparison with tree clutter spectral results indicates that, when an occasional water body comes under surveillance at vertical polarization in otherwise generally forested terrain, water clutter spectral density is expected to exceed surrounding-terrain tree clutter spectral densities in the Bragg-offset Doppler vicinity by large amounts     

Multitarget Bayes filtering via first-order multitarget moments

The theoretically optimal approach to multisensor-multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. Even in single-target problems, this optimal filter is so computationally challenging that it must usually be approximated. Consequently, multitarget Bayes filtering will never be of practical interest without the development of drastic but principled approximation strategies. In single-target problems, the computationally fastest approximate filtering approach is the constant-gain Kalman filter. This filter propagates a first-order statistical moment - the posterior expectation - in the place of the posterior distribution. The purpose of this paper is to propose an analogous strategy for multitarget systems: propagation of a first-order statistical moment of the multitarget posterior. This moment, the probability hypothesis density (PHD), is the function whose integral in any region of state space is the expected number of targets in that region. We derive recursive Bayes filter equations for the PHD that account for multiple sensors, nonconstant probability of detection, Poisson false alarms, and appearance, spawning, and disappearance of targets. We also show that the PHD is a best-fit approximation of the multitarget posterior in an information-theoretic sense.     

A Comparison of Complementary and Kalman Filtering

A technique used in the flight control industry for estimation when combining measurements is the complementary filter. This filter is usually designed without any reference to Wiener or Kalman filters, although it is related to them. This paper, which is mainly tutorial, reviews complementary filtering and shows its relationship to Kalman and Wiener filtering.     

Correlation between horizontal and vertical monopulse measurements

Many radar systems use the monopulse ratio to extract angle of arrival (AOA) measurements in both azimuth and elevation angles. The accuracies of each such measurement are reasonably well known: each measurement is, conditioned on the sum-signal return, Gaussian-distributed with calculable bias (relative to the true AOA), and variance. However, we note that the two monopulse ratios are functions of basic radar measurements that are not entirely independent, specifically in that the sum signal is common to both. The effect of this is that the monopulse ratios are dependent, and a simple explicit expression is given for their correlation; this is of considerable interest when the measurements are supplied to a tracking algorithm that requires a measurement covariance matrix. The system performance improvement when this is taken into account is quantified: while it makes little difference for a tracking radar with small pointing errors, there are more substantial gains when a target is allowed to stray within the beam, as with a rotating (track-while-scan) radar or when a single radar dwell interrogates two or more targets at different ranges. But in any case, the correct covariance expression is so simple that there is little reason not to use it. We additionally derive the Cramer-Rao lower bound (CRLB) on joint azimuth/elevation angle estimation and discover that it differs only slightly from the covariance matrix corresponding to the individual monopulse ratios. Hence, using the individual monopulse ratios and their simple joint accuracy expression is an adequate and quick approximation of the optimal maximum likelihood procedure for single resolved targets.     

Resonant Cavity Measurements of Ionized Wakes

A transmission resonant cavity technique, which is suitable for making measurements of electron line densities and collision frequencies in the ionized wakes of hypervelocity projectiles, is described. With this method electron density measurements can be made over six orders of magnitude. Resonant cavity design requirements and limitations of the method are discussed. Typical data from measurements behind projectiles traveling at speeds up to 6.5 km/s are given.     

Bearings-only and Doppler-bearing tracking using instrumentalvariables

In bearings-only tracking (BOT) or Doppler and bearing tracking (DBT), both common passive sonar problems, the measurement equations are nonlinear. To apply the Kalman filter, it is necessary either to linearize the equations or to embed the nonlinearities into the noise terms. The former sometimes leads to filter divergence, while the latter produces biased estimates. A formulation of BOT and DBT which has a constant state vector and simplifies the tracking problem to one of constant parameter estimation is given. The solution is by the instrumental variable method. The instrumental variables are obtained from predictions based on past measurements and are therefore independent of the present noisy measurements. The result is a recursive unbiased estimator. The theoretical developments are verified by simulation, which also shows that the formulation leads to near optimal estimators whose errors are close to the Cramer-Rao lower bound (CRLB)     

复合正交级数神经网络模型在飞控中的应用

针对现有神经网络逼近快变非线性函数能力低、结构复杂等问题,提出了复合正交多项式模型和一种改进的正匀多项式神经网络模型。这两种模型综合了正交级数和神经网络的结构特点,便于工程实现。理论研究和仿真计算表明,使用此模型构造快变非线性函数时,可使神经网络的节点数成倍减少,神经网络的训练次数成倍降低。飞行器非线性控制的应用表现,此方法可用来解决一般非线性系统的控制问题。     

一种非线性滤波器的误差分析及应用

针对一类具有未建模误差和扰动的非线性系统的状态估计问题,提出一种在线估计并补偿模型误差的非线性滤波算法,该算法利用非线性预测滤波(NPF)基于预测输出残差的方差最小的基本原则估计模型误差,冉利用扩展卡尔曼滤波(EKF)的思想对补偿后的模型进行状态估计;详细推导了状态估计误差及其方差阵的传播模型.以卫星姿态确定系统为例,...     

利用差值定理降低飞行器速度和加速度拟合的截断误差

推导了差值定理及其推论,并以此作为数据拟合的有效点判据,显著降低了飞行器速度和加速度拟合的截断误差;探讨了该判据条件下滤波参数对拟合结果的影响及截断误差的判断条件,并采用实测数据对拟合效果进行了检验。     

State estimation using an approximate reduced statistics algorithm

The problem of state estimation using nonlinear additive Gaussian noise measurements is addressed. A geometric model for the posterior state density is assumed based on a multidimensional Haar basis representation. An approximate reduced statistics (ARS) algorithm, suggested by the parameter estimator of Kulhavy is then developed, using successive minimization of relative entropy between model densities and an approximate posterior density. The state estimator thus derived is applied to a bearings-only target tracking problem in a multiple sensor scenario     

自适应滤波方法研究

证明Ｓａｇｅ－Ｈｕｓａ的自适应卡尔曼滤波算法不能同时估计Ｑ和Ｒ,并分析了该法导致滤波发散的原因。介绍了一种新的自适应卡尔曼滤波算法,该法当Ｒ（或Ｑ）已知时可以准确地估计出Ｑ（或Ｒ）。该法的独特之处在于当对Ｑ（或Ｒ）进行修正估计时,只采用矩阵的乘运算和求逆运算,而不进行加减法运算,因此消除了滤波发散现象。数字仿真表明效果良好。     

An Adaptive Two-Dimensional Kalman Tracking Filter

A continuously adaptive two-dimensional Kalman tracking filter for a low data rate track-while-scan (TWS) operation is introduced which enhances the tracking of maneuvering targets. The track residuals in each coordinate, which are a measure of track quality, are sensed, normalized to unity variance, and then filtered in a single-pole filter. The magnitude Z of the output of this single-pole filter, when it exceeds a threshold Z1 is used to vary the maneuver noise spectral density q in the Kalman filter model in a continuous manner. This has the effect of increasing the tracking filter gains and containing the bias developed by the tracker due to the maneuvering target. The probability of maintaining track, with reasonably sized target gates, is thus increased, The operational characteristic of q versus Z assures that the tracker gains do not change unless there is high confidence that a maneuver is in progress.