A Kalman Tracker with a Simple Input Estimator

Two Kalman filter based schemes are proposed for tracking maneuvering targets. Both schemes use least squares to estimate a target's acceleration input vector and to update the tracker by this estimate. The first scheme is simpler and by an approximation to its input estimator the computation can be considerably reduced with insignificant performance degradation. The second scheme requires two Kalman filters and hence is more complex. However, since one of its two filters assumes input noise, it may outperform the first scheme when input noise is indeed present. A detector that compares the weighted norm of the estimated input vector to a threshold is used in each scheme. Its function is to guard against false updating of the trackers and to keep the error covariance small during constant velocity tracks. Simulation results for various target profiles are included. They show that in terms of tracking performance, both schemes are comparable. However, because of its computation simplicity, the first scheme is far superior.     

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.     

Discrete-time min-max tracking

The problem of optimal state estimation of linear discrete-time systems with measured outputs that are corrupted by additive white noise is addressed. Such estimation is often encountered in problems of target tracking where the target dynamics is driven by finite energy signals, whereas the measurement noise is approximated by white noise. The relevant cost function for such tracking problems is the expected value of the standard H/sub /spl infin// performance index, with respect to the measurement noise statistics. The estimator, serving as a tracking filter, tries to minimize the mean-square estimation error, and the exogenous disturbance, which may represent the target maneuvers, tries to maximize this error while being penalized for its energy. The solution, which is obtained by completing the cost function to squares, is shown to satisfy also the matrix version of the maximum principle. The solution is derived in terms of two coupled Riccati difference equations from which the filter gains are derived. In the case where an infinite penalty is imposed on the energy of the exogenous disturbance, the celebrated discrete-time Kalman filter is recovered. A local iterations scheme which is based on linear matrix inequalities is proposed to solve these equations. An illustrative example is given where the velocity of a maneuvering target has to be estimated utilizing noisy measurements of the target position.     

Application of the Kalman-Levy Filter for Tracking Maneuvering Targets

Among target tracking algorithms using Kalman filtering-like approaches, the standard assumptions are Gaussian process and measurement noise models. Based on these assumptions, the Kalman filter is widely used in single or multiple filter versions (e.g., in an interacting multiple model (IMM) estimator). The oversimplification resulting from the above assumptions can cause degradation in tracking performance. In this paper we explore the application of Kalman-Levy filter to handle maneuvering targets. This filter assumes a heavy-tailed noise distribution known as the Levy distribution. Due to the heavy-tailed nature of the assumed distribution, the Kalman-Levy filter is more effective in the presence of large errors that can occur, for example, due to the onset of acceleration or deceleration. However, for the same reason, the performance of the Kalman-Levy filter in the nonmaneuvering portion of track is worse than that of a Kalman filter. For this reason, an IMM with one Kalman and one Kalman-Levy module is developed here. Also, the superiority of the IMM with Kalman-Levy module over only Kalman-filter-based IMM for realistic maneuvers is shown by simulation results.     

机动目标“当前”统计模型与自适应跟踪算法

本文提出机动目标“当前”统计模型的概念并建议用修正的瑞利-马尔科夫过程描述目标随机加速机动的统计特性。文中指出了在机动目标运动模型中状态（机动加速度）估值与状态噪声之间的内在联系。在此基础上提出了具有机动加速度均值及方差自适应的卡尔曼滤波算法。对一维和三维的情形进行了计算机模拟。计算结果表明,在仅对目标位置进行观测的情况下,这类自适应估值算法无论对高度机动或无机动的目标均可绘出较好的位置、速度及加速度估值。     

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.     

Canonical transform for tracking with kinematic models

A canonical transform is presented that converts a coupled or uncoupled kinematic model for target tracking into a decoupled dimensionless canonical form. The coupling is due to non-zero off-diagonal terms in the covariance matrices of the process noise and/or the measurement noise, which can be used to model the coupling of motion and/or measurement between coordinates. The decoupled dimensionless canonical form is obtained by simultaneously diagonalizing the noise covariance matrices, followed by a spatial-temporal normalization procedure. This canonical form is independent of the physical specifications of an actual system. Each subsystem corresponding to a canonical coordinate is characterized by its process noise standard deviation, called the maneuver index as a generalization of the tracking index for target tracking, which characterizes completely the performance of a steady-state Kalman filter. A number of applications of this canonical form are discussed. The usefulness of the canonical transform is illustrated via an example of performance analysis of maneuvering target tracking in an air traffic control (ATC) system.     

基于IMMKF算法的ADS-B监视应用目标跟踪

目标跟踪是机载广播式自动相关监视（ADS-B）应用的基础功能,对提升航空器周边的弱机动民航飞机目标跟踪性能具有重要意义。提出一种基于交互式多模型卡尔曼滤波（IMMKF）算法的ADS-B 监视应用目标跟踪方法。首先,针对弱机动背景下的民航飞机的飞行特点,建立包含匀速模型和标准协同转弯模型的运动模型集,并对模型进行线性化近似;然后,将模型预测和ADS-B 状态矢量量测数据作为IMMKF 算法中多个并行卡尔曼滤波器的输入,进行并行滤波;最后,计算得到目标状态矢量的估计和模型近似概率,并作为下一次迭代的输入。结果表明：相比于基于匀速模型的卡尔曼滤波目标跟踪方法,IMMKF 方法的位置跟踪误差降低了59%,速度跟踪误差降低了77%,显著提升了状态估计性能,具备较高的跟踪精度、稳健性与计算效率,在ADS-B 监视应用中具有实际应用价值与借鉴意义。     

机动多目标跟踪问题中关联区域的研究

 本文研究了跟踪多个机动目标时,由滤波算法所获得的新息向量范数的统计性质,关联区域的大小以及接收正确回波的概率。借助拉蒙特卡洛方法,考察了不同的目标状态模型、目标机动加速度及状态噪声方差等因素对所研究的问题的影响。研究表明,文献[1]所提出的机动目标状态模型及相应的自适应算法具有较好的适应目标机动的能力,关联区域的大小及接收正确回波的概率均较为稳定。     

Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets

The majority of tactical weapons systems require that manned maneuverable vehicles, such as aircraft, ships, and submarines, be tracked accurately. An optimal Kalman filter has been derived for this purpose using a target model that is simple to implement and that represents closely the motions of maneuvering targets. Using this filter, parametric tracking accuracy data have been generated as a function of target maneuver characteristics, sensor observation noise, and data rate and that permits rapid a priori estimates of tracking performance to be made when maneuvering targets are to be tracked by sensors providing any combination of range, bearing, and elevation measurements.     

Reduced state estimators for consistent tracking of maneuvering targets

Linear Kalman filters, using fewer states than required to completely specify target maneuvers, are commonly used to track maneuvering targets. Such reduced state Kalman filters have also been used as component filters of interacting multiple model (IMM) estimators. These reduced state Kalman filters rely on white plant noise to compensate for not knowing the maneuver - they are not necessarily optimal reduced state estimators nor are they necessarily consistent. To be consistent, the state estimation and innovation covariances must include the actual errors during a maneuver. Blair and Bar-Shalom have shown an example where a linear Kalman filter used as an inconsistent reduced state estimator paradoxically yields worse errors with multisensor tracking than with single sensor tracking. We provide examples showing multiple facets of Kalman filter and IMM inconsistency when tracking maneuvering targets with single and multiple sensors. An optimal reduced state estimator derived in previous work resolves the consistency issues of linear Kalman filters and IMM estimators.     

Tracking targets using adaptive Kalman filtering

A simple algorithm for estimating the unknown process noise variance of an otherwise known linear plant, using a Kalman filter is suggested. The process noise variance estimator is essentially dead beat, using the difference between the expected prediction error variance, computed in the Kalman filter, and the measured prediction error variance. The estimate is used to adapt the Kalman filter. The use of the adaptive filter is demonstrated in a simulated example in which a wildly maneuvering target is tracked     

A three-state Kalman tracker using position and rate measurements

A three-state Kalman tracker is described for tracking a moving target, such as an aircraft, making use of the position and rate measurements obtained by a track-white-scan radar sensor which employs pulsed Doppler processing, such as the moving target detector providing unambiguous Doppler data. The steady-state filter parameters have been analytically obtained under the assumption of white noise maneuver capability. The numerical computations of these parameters are in excellent agreement with those obtained from the recursive Kalman filter matrix equations. The solution for the case when only the range measurements are available is obtained as a special case of this model. Graphs of normalized covariances and gains are presented to illustrate how the solution depends on different parameters     

Adaptive maneuvering entry guidance with ground-track control

We demonstrate that a three-dimensional entry guidance with accurate ground-track control can be achieved using a combined strategy of proportional down-range maneuvering and linear quadratic (LQ) technique, thereby precluding the necessity of sophisticated nonlinear programming. It is also shown that heating rate and load-factor constraints can be met by a careful but simple selection of LQ gains. Although the speed, flight path angle, and down-range accuracy are comparable to that of a planar guidance strategy, the present technique also demonstrates an accurate cross-range control, which is not possible with a planar strategy. It is shown that adaptive maneuvering produces a superior performance than that of a nonlinear tracking control derived using feedback linearization. The method can handle much larger deviations in the entry conditions than those published previously, and has the important advantage of a relatively simple implementation in a realistic entry scenario. Furthermore, the LQ technique can take advantage of the good robustness properties of a linear Kalman filter based upon the measurement of longitude, latitude, and speed, wherein little deterioration is observed in the closed-loop performance.     

一种基于加速度预估计的机动目标跟踪算法

研究分析了几种典型单机动目标模型的建模方法,针对现有单机动目标模型中机动参数需要先验假设,并且不能随目标机动情况的改变而自适应调整的问题,提出了一种加速度预估计模型(Acceleration Pre-estimation Model,APM)。该模型首先用位置量测对机动加速度进行预估计;然后,将加速度估计值作为系统的输入控制项建模;将估计误差看做系统的机动控制项,并作为系统的相关噪声建模。由于APM模型中,加速度机动参数是通过位置量测实时估计得到的,不需要先验假设。与现有单机动目标模型相比,该模型的自适应能力得到了提高。     

Frequency estimation techniques for high dynamic trajectories

A comparison is presented of four different estimation techniques applied to the problem of continuously estimating the rapidly varying parameters of a sinusoidal signal, observed in the presence of additive noise. Frequency estimates are emphasized, although phase and/or frequency rate are also estimated by some of the algorithms. These parameters are related to the velocity, position, and acceleration of the maneuvering receiver or transmitter. Estimated performance at low carrier-to-noise ratios and high dynamics is investigated for the purpose of determining the useful operating range of an approximate maximum likelihood estimator, an extended Kalman filter, a cross-product automatic frequency loop and a phase-locked loop. Numerical simulations are used to evaluate performance while tracking a common trajectory exhibiting high dynamics     

A Kalman Filter Based Tracking Scheme with Input Estimation

Beginning with the derivation of a least squares estimator that yields an estimate of the acceleration input vector, this paper first develops a detector for sensing target maneuvers and then develops the combination of the estimator, detector, and a "simple" Kalman filter to form a tracker for maneuvering targets. Finally, some simulation results are presented. A relationship between the actual residuals, assuming target maneuvers, and the theoretical residuals of the "simple" Kalman filter that assumes no maneuvers, is first formulated. The estimator then computes a constant acceleration input vector that best fits that relationship. The result is a least squares estimator of the input vector which can be used to update the "simple" Kalman filter. Since typical targets spend considerable periods of time in the constant course and speed mode, a detector is used to guard against automatic updating of the "simple" Kalman filter. A maneuver is declared, and updating performed, only if the norm of the estimated input vector exceeds a threshold. The tracking sclheme is easy to implement and its capability is illustrated in three tracking examples.     

Analytical solution of discrete colored noise ECA tracking filter

New analytical solutions of steady-state Kalman gains are presented for a discrete-time tracking filter with correlation in both the measurement noise and the target maneuver. The measurement noise model is a first-order discrete Markov process characterized by a correlation coefficient ρ. The target motion is examined for an exponentially correlated acceleration maneuver type in which the vehicle oscillation such as wind-induced-bending is also considered. The present solution method is based on factorizing the observed spectral density matrix Ψ(z) in frequency domain. The algorithm proposed here gives the Kalman gain matrix directly. For a case when the steady-state error covariance matrix is desired, such gains can be incorporated with the algebraic Riccati equation     

基于模糊自适应卡尔曼滤波的大气数据辅助姿态算法

 针对中低精度航姿参考系统(AHRS)在机体机动时不能利用加速度计修正水平姿态,以及噪声统计特性随实际工作情况变化的问题,提出了一种基于模糊自适应卡尔曼滤波的大气数据辅助姿态解算的方法。首先,考虑大气数据系统和航姿参考系统的优势,利用真空速、攻角和侧滑角等大气数据信息对非重力加速度进行补偿,以辅助水平姿态解算;其次,基于模糊自适应卡尔曼滤波原理,对观测模型的参数进行估计和修正,以实现水平姿态的最优估计;最后,选取某型飞机的试飞数据进行仿真验证。仿真结果表明,该方法可使飞机的水平姿态估计精度达到1.3°,且在偏差较大时有明显的纠偏作用。因此,相对于无机动加速度补偿和常规卡尔曼滤波来说,该方法能够更好地进行姿态估计,具有一定的实用价值。     

一类新型机动目标跟踪算法

阐述了当跟踪非机动目标时,传统的Kalman滤波可以得到很好的跟踪精度。但是当日标机动时,传统的Kalman滤波不能对目标的突然变化做出及时的改正和预测,因此跟踪精度很差,甚至出现丢失目标的情况。文中采用的基于截断正态概率模型的改进自适应目标跟踪算法, 其结构和计算简单,鲁棒性好,较好地解决了使用Kalman滤波带来的不足。