Transient Response of Tracking Filters with Randomly Interrupted Data

The transient responses during the initialization phase of a first-order ?-? tracking filter and a second-order Kalman filter are evaluated as a function of radar measurement accuracy and the probability of receiving valid data at the prescribed intervals. Monte Carlo simulation results are complemented by analysis of the filtering processes and curves are presented which clearly define the deterioration in filter performance attributable to reduced probabilities of data acquisition. In addition, the responses of ?-? and Kalman filters are shown to be identical when the ?, ? gains are selected optimally.     

On a decoupled multitarget tracking algorithm

An analysis is conducted of the optimality of a decoupled tracking filtering algorithm for addressing the problem of tracking multiple targets with correlated measurements and maneuvers. It is proved that the decoupled filters are, in general, suboptimal and are not in fact Kalman filters. However, it is shown also that if the standard Kalman filter is asymptotically stable, the decoupled filters will converge asymptotically to the stable version of the standard Kalman filter. For the case of time-invariant measurement and process noise covariance, a simple sufficient condition guaranteeing the asymptotical stability of the decoupled filters are given     

Adaptive Tracking Filter for Maneuvering Targets

A general method of continually restructuring an optimum Bayes-Kalman tracking filter is proposed by conceptualizing a growing tree of filters to maintain optimality on a target exhibiting maneuver variables. This tree concept is then constrained from growth by quantizing the continuously sensed maneuver variables and restricting these to a small value from which an average maneuver is calculated. Kalman filters are calculated and carried in parallel for each quantized variable. This constrained tree of several parallel Kalman filters demands only modest om; puter time, yet provides very good performance. This concept is implemented for a Doppler tracking system and the performance is compared to an extended Kalman filter. Simulation results are presented which show dramatic tracking improvement when using the adaptive tracking filter.     

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.     

一种基于二级变维Kalman滤波的跟踪算法

Kalman滤波器是一种高速的目标跟踪器.针对不同阶数的Kalman滤波器具有不同的跟踪能力与跟踪效率之间存在的矛盾,设计了一种自适应Kalman滤波算法.该算法使用两级滤波器,根据目标机动性的变化,适当的调整滤波器的阶数,使跟踪结果快速收敛,很好地解决了矛盾.通过对仿真结果分析表明,算法具有可靠、计算简便、快速等特点,模型滤波精度较高,并可实现实时跟踪预测,具有一定的理论价值和实用价值.     

一种新的基于机动检测的机动目标跟踪算法

针对Kalman滤波跟踪机动目标发散和目前多数自适应Kalman滤波算法对运动模型适应性不强的问题,提出了一种新的基于机动检测的机动目标跟踪算法,通过实时自适应的改变滤波模型提高对机动目标跟踪精度。对这种方法与Kalman滤波算法进行了计算机仿真比较,结果表明,该方法计算量小,可实时精确地自适应匹配目标的运动模型,可实现对机动目标稳定可靠的跟踪。     

Kalman filtering with state equality constraints

Kalman filters are commonly used to estimate the states of a dynamic system. However, in the application of Kalman filters there is often known model or signal information that is either ignored or dealt with heuristically. For instance, constraints on state values (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. A rigorous analytic method of incorporating state equality constraints in the Kalman filter is developed. The constraints may be time varying. At each time step the unconstrained Kalman filter solution is projected onto the state constraint surface. This significantly improves the prediction accuracy of the filter. The use of this algorithm is demonstrated on a simple nonlinear vehicle tracking problem     

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.     

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.     

自适应高阶容积卡尔曼滤波在目标跟踪中的应用

针对传统容积卡尔曼滤波(CKF)在系统状态发生突变时估计精度下降的问题,将强跟踪滤波(STF)算法与高阶容积卡尔曼滤波(HCKF)算法相结合,提出了一种自适应高阶容积卡尔曼滤波(AHCKF)方法。该算法采用高阶球面-相径容积规则,可获得高于传统CKF的估计精度,同时在HCKF算法中引入STF,通过渐消因子在线修正预测误差协方差阵,强迫残差序列正交,提高了算法的鲁棒性,增强了算法应对系统状态突变等不确定因素的能力。将提出的AHCKF算法应用于具有状态突变的机动目标跟踪问题并进行数值仿真,仿真结果表明,AHCKF算法在系统状态发生突变的情况下表现出良好的滤波性能,有效地避免了状态突变造成的滤波精度下降,较传统的CKF、HCKF、交互式多模型-容积滤波(IMM-CKF)和自适应容积卡尔曼滤波(ACKF)算法有更强的鲁棒性和系统自适应能力。     

Maneuvering target tracking using extended Kalman filter

A numerically well-conditioned, quasi-extended Kalman filter is proposed. The filter is numerically described. The simulation results presented show that the estimation performance of the quasi-extended filter is superior, for short distances, compared with the widely used linear tracking filters. In addition, the simplicity of the quasi-extended filter makes it very easy to implement     

Digital tracking filters with high order correlated measurement noise

The existing algorithms for the design of digital filters with colored measurement noise involve a restriction on the dimension of the measurement error model. Kalman filter equations and state space partition are used to formulate an optimal tracking filter without such restrictions. The input to the new filter are two consecutive measurements, and it is initialized by using the first available measurements and the error model correlation matrix. Several examples illustrate the filter formulation and initialization.     

Incremental maneuver estimation model for target tracking

An incremental model for maneuver detection and estimation for use in target tracking with the Kalman filter is described. The approach is similar to the multiple Kalman filter bank, but with a memory for the maneuver status for the track under consideration. The advantage of this approach is that the target acceleration can be more accurately estimated. The maneuver-detection model has shown good maneuver-following capability. Moreover, it needs only a finite number of Kalman filters to handle all possible maneuver values and it responds quickly as maneuver occurs. When there is an abrupt maneuver change the model can still track the targets in short time     

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。     

Target tracking in maneuver-centered coordinates

Aircraft targets normally maneuver on circular paths, which has led to tracking filters based on circular turns. A coordinate system to track circular maneuvers with a simple Kalman filter is introduced. This system is a polar coordinate system located at the center of the maneuver. It leads to a tracking filter with range, angle, and angular velocity in the state vector. Simulation results are presented, showing that the algorithm displays improved performance over methods based on constant x-y acceleration when tracking circular turns     

自由段弹道目标跟踪算法在工程中的应用

在对弹道目标跟踪预警的工程实践中,雷达系统对目标运动的信息处理速度尤为重要,因而,文章选取自适应跟踪模型与卡尔曼滤波相结合的方法解决自由段弹道目标的跟踪问题,并与扩展卡尔曼跟踪算法做了对比分析。仿真显示,2种滤波方式分别与自适应跟踪模型相结合后,卡尔曼滤波和扩展卡尔曼滤波跟踪性能相差不大,但其算法简单、运算时间短,可以较好满足自由段弹道目标跟踪的工程需求。     

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.     

The Tracking Index: A Generalized Parameter for ? ? ? and ? ? ? ? Target Trackers

A generalized, optimal filtering solution is presented for the target tracking problem. Applying optimal filtering theory to the target tracking problem, the tracking index, a generalized parameter proportional to the ratio of the position uncertainty due to the target maneuverability to that due to the sensor measurement, is found to have a fundamental role not only in the optimal steady-state solution of the stochastic regulation tracking problem, but also in the track initiation process. Depending on the order of the tracking model, the tracking index solution yields a closed form, consistent set of generalized tracking gains, relationships, and performances. Using the tracking index parameter, an initializing and tracking procedure in recursive form, realizes the accuracy of the Kalman filter with an algorithm as simple as the well-known ? ? ? filter or ? ? ? ? ? filter depending on the tracking order.     

Information type filters for tracking a maneuvering target

Two previously proposed adaptive covariance-type Kalman filtering techniques for tracking maneuvering targets (see Y.T. Chan et al. ibid., p.237-44, Mar. 1979, and Z. Tang et al. Report, Department of Electrical and Computer Engineering, Oregon State University, Corvallis, Oct. 1983) are developed further to utilize the information-type Kalman filter. These adaptive information-type filters are described in structurally decoupled forms, thereby greatly reducing the computational requirements while rendering the filters amenable to implementation on parallel processors. A coherent decision procedure for including partial coupling when necessary is developed via offline analysis of crosscorrelation functions     

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.