Track-to-track fusion with dissimilar sensors

An analysis is described of a kinematic state vector fusion algorithm when tracks are obtained from dissimilar sensors. For the sake of simplicity, it is assumed that two dissimilar sensors are equipped with nonidentical two-dimensional optimal linear Kalman filters. It is shown that the performance of such a track-to-track fusion algorithm can be improved if the cross-correlation matrix between candidate tracks is positive. This cross-correlation is introduced by noise associated with target maneuver that is common to the tracking filters in both sensors and is often neglected. An expression for the steady state cross-correlation matrix in closed form is derived and conditions for positivity of the cross-correlation matrix are obtained. The effect of positivity on performance of kinematic track-to-track fusion is also discussed     

Performance evaluation of track fusion with information matrixfilter

In a multisensor environment, each sensor detects multiple targets and creates corresponding tracks. Fusion of tracks from these, possibly dissimilar, sensors yields more accurate kinematic and attribute information regarding the target. Two methodologies have been employed for such purpose, which are: measurement fusion and state vector fusion. It is well known that the measurement fusion approach is optimal but computationally inefficient and the state vector fusion algorithms are more efficient but suboptimal, in general. This is so because the state vector estimates to be fused obtained from two sensors, are not conditionally independent in general due to the common process noise from the target being tracked. It is to be noted that there are three approaches to state vector fusion, which are: weighted covariance, information matrix, and pseudomeasurement. This research is restricted solely to performance evaluation of the information matrix form of state vector fusion. Closed-form analytical solution of steady state fused covariance has been derived as a measure of performance using this approach. Note that the results are derived under the assumptions that the two sensors are synchronized and no misassociation or merged measurement is considered in the study. Results are compared with those using Monte Carlo simulation, which was used in the past to predict fusion system performance by various authors. These results provide additional insight into the mechanism of track fusion and greatly simplify evaluation of fusion performance. In addition, availability of such a solution facilitates the trade-off studies for designing fusion systems under various operating conditions     

Performance evaluation for MAP state estimate fusion

This paper presents a quantitative performance evaluation method for the maximum a posteriori (MAP) state estimate fusion algorithm. Under ideal conditions where data association is assumed to be perfect, it has been shown that the MAP or best linear unbiased estimate (BLUE) fusion formula provides the best linear minimum mean squared estimate (LMMSE) given local estimates under the linear Gaussian assumption for a static system. However, for a dynamic system where fusion is recursively performed by the fusion center on local estimates generated from local measurements, it is not obvious how the MAP algorithm will perform. In the past, several performance evaluation methods have been proposed for various fusion algorithms, including simple convex combination, cross-covariance combination, information matrix, and MAP fusion. However, not much has been done to quantify the steady state behavior of these fusion methods for a dynamic system. The goal of this work is to present analytical fusion performance results for MAP state estimate fusion without extensive Monte Carlo simulations, using an approach developed for steady state performance evaluation for track fusion. Two different communication strategies are considered: fusion with and without feedback to the sensors. Analytic curves for the steady state performance of the fusion algorithm for various communication patterns are presented under different operating conditions.     

Architectures and algorithms for track association and fusion

Target tracking using multiple sensors can provide better performance than using a single sensor. One approach to multiple target tracking with multiple sensors is to first perform single sensor tracking and then fuse the tracks from the different sensors. Two processing architectures for track fusion are presented: sensor to sensor track fusion, and sensor to system track fusion. Technical issues related to the statistical correlation between track estimation errors are discussed. Approaches for associating the tracks and combining the track state estimates of associated tracks that account for this correlation are described and compared by both theoretical analysis and Monte Carlo simulations     

On optimal track-to-track fusion

Track-to-track fusion is an important part in multisensor fusion. Much research has been done in this area. Chong et al. (1979, 1986, 1990) among others, presented an optimal fusion formula under an arbitrary communication pattern. This formula is optimal when the underlying systems are deterministic, i.e., the process noise is zero, or when full-rate communication (two sensors exchange information each time they receive new measurements) is employed. However, in practice, the process noise is not negligible due to target maneuvering and sensors typically communicate infrequently to save communication bandwidth. In such situations, the measurements from two sensors are not conditionally (given the previous target state) independent due to the common process noise from the underlying system, and the fusion formula becomes an approximate one. This dependence phenomena was also observed by Bar-Shalom (1981) where a formula was derived to compute the cross-covariance of two track estimates obtained by different sensors. Based on this results a fusion formula was subsequently derived (1986) to combine the local estimates which took into account the dependency between the two estimates. Unfortunately, the Bayesian derivation made an assumption that is not met. This work points out the implicit approximation made and shows that the result turns out to be optimal only in the ML (maximum likelihood) sense. A performance evaluation technique is then proposed to study the performance of various track-to-track fusion techniques. The results provide performance bounds of different techniques under various operating conditions which can be used in designing a fusion system.     

Efficient recursive state estimator for dynamic systems withoutknowledge of noise covariances

An efficient recursive state estimator for dynamic systems without knowledge of noise covariances is suggested. The basic idea for this estimator is to incorporate the dynamic matrix and the forgetting factor into the least squares (LS) method to remedy the lack of knowledge of noises. We call it the extended forgetting factor recursive least squares (EFRLS) estimator. This estimator is shown to have similar asymptotic properties to a completely specified Kalman filter state estimator. More importantly, the performance of EFRLS greatly exceeds that of existing filtering techniques when the noise variance is misspecified. In addition, EFRLS also performs well when there is cross-correlation between the process and measurement noise streams or temporal dependencies within those streams. Some discussions and a number of simulations are made to provide practical guidance on the choice of an optimal forgetting factor and evaluate the performance of the EFRLS algorithms, which strongly dominates that of the standard forgetting factor recursive least squares (FRLS) and some misspecified Kalman filtering     

Performance evaluation of multi-sensor classification systems

A common problem in classification is to use one/more sensors to observe repeated measurements of a target's features/attributes, and in turn update the targets' posterior classification probabilities to aid in target identification. This paper addresses the following questions: 1. How do we quantify the classification performance of a sensor? 2. What happens to the posterior probabilities as the number of measurements increase? 3. Will the targets be classified correctly? While the Kalman filter allows for off-line estimation of kinematic performance (covariance matrix), a comparable approach for studying classification accuracy has not been done previously. We develop a new analytical approach for computing the long-run classification performance of a sensor and also present recursive formulas for efficient calculation of the same. We show that, under a minimal condition, a sensor will eventually classify all targets perfectly. We also develop a methodology for evaluating the classification performance of multi-sensor fusion systems involving sensors of varying quality. The contributions of this paper are 1. A simple metric to quantify a sensor's ability to discriminate between the targets being identified, and its use in comparing multiple sensors, 2. An approximate formula based on this metric to compute off-line estimates of the rate of convergence toward perfect classification, and the number of measurements required to achieve a desired level of classification accuracy, and 3. The use of this metric to evaluate classification performance of multi-sensor fusion systems.     

圆锥低超声速有攻角绕流非对称涡流的数值模拟与非线性分岔的研究

本文用数值方法研究了圆锥低超声速有攻角绕流的对称和非对称定常解,扰动响应以及在更大角时出现的准周期解问题。     

Lightweight hybrid visual-inertial odometry with closed-form zero velocity update

Visual-Inertial Odometry (VIO) fuses measurements from camera and Inertial Measurement Unit (IMU) to achieve accumulative performance that is better than using individual sensors. Hybrid VIO is an extended Kalman filter-based solution which augments features with long tracking length into the state vector of Multi-State Constraint Kalman Filter (MSCKF). In this paper, a novel hybrid VIO is proposed, which focuses on utilizing low-cost sensors while also considering both the computational efficiency and positioning precision. The proposed algorithm introduces several novel contributions. Firstly, by deducing an analytical error transition equation, one-dimensional inverse depth parametrization is utilized to parametrize the augmented feature state. This modification is shown to significantly improve the computational efficiency and numerical robustness, as a result achieving higher precision. Secondly, for better handling of the static scene, a novel closed-form Zero velocity UPdaTe (ZUPT) method is proposed. ZUPT is modeled as a measurement update for the filter rather than forbidding propagation roughly, which has the advantage of correcting the overall state through correlation in the filter covariance matrix. Furthermore, online spatial and temporal calibration is also incorporated. Experiments are conducted on both public dataset and real data. The results demonstrate the effectiveness of the proposed solution by showing that its performance is better than the baseline and the state-of-the-art algorithms in terms of both efficiency and precision. A related software is open-sourced to benefit the community.^①     

Comparison of two-sensor tracking methods based on state vectorfusion and measurement fusion

There are two approaches to the two-sensor track-fusion problem. Y Bar-Shalom and L. Campo (ibid., vol.AES-22, 803-5, Nov. 1986) presented the state vector fusion method, which combines state vectors from the two sensors to form a new estimate while taking into account the correlated process noise. The measurement fusion method or data compression of D. Willner et al. (1976) combines the measurements from the two sensors first and then uses this fused measurement to estimate the state vector. The two methods are compared and an example shows the amount of improvement in the uncertainty of the resulting estimate of the state vector with the measurement fusion method     

Exact multisensor dynamic bias estimation with local tracks

An exact solution is provided for the multiple sensor bias estimation problem based on local tracks. It is shown that the sensor bias estimates can be obtained dynamically using the outputs of the local (biased) state estimators. This is accomplished by manipulating the local state estimates such that they yield pseudomeasurements of the sensor biases with additive noises that are zero-mean, white, and with easily calculated covariances. These results allow evaluation of the Cramer-Rao lower bound (CRLB) on the covariance of the sensor bias estimates, i.e., a quantification of the available information about the sensor biases in any scenario. Monte Carlo simulations show that this method has significant improvement in performance with reduced rms errors of 70% compared with commonly used decoupled Kalman filter. Furthermore, the new method is shown to be statistically efficient, i.e., it meets the CRLB. The extension of the new technique for dynamically varying sensor biases is also presented.     

基于PTE的多雷达航迹融合算法

针对杂波背景下的多雷达航迹融合时局部估计误差互协方差矩阵未知的问题,提出基于目标存在概率（PTE）的航迹融合算法,提升了正确航迹率和跟踪精度。首先,通过综合概率数据关联得到单接收站的目标航迹估计集合和对应的目标存在概率。然后,在局部估计误差互协方差矩阵未知的条件下,基于PTE信息提出不带记忆的综合广义凸组合航迹融合算法。进而将前一帧的融合状态进行反馈,提出带记忆的综合广义凸组合航迹融合算法。仿真验证了所提算法的有效性。     

基于灰色理论的一种数据融合方法

在灰色关联分析的基础上,对斜关联度进行了修正,引出了点、斜修正关联度分析的概念.通过对影响目标属性识别的各种因素进行分析,结合战术思想利用灰色点、斜修正关联度分析及多目标优化方法建立了数据融合模型,提出了一种基于灰色理论的多传感器数据融合方法.计算多传感器测量数据的灰色关联矩阵,进行灰色优势分析,然后进行数据融合.此方法考虑了各传感器测量数据的精确度,而且删除掉了测量比较差或测量不到的数据.仿真结果表明,应用该方法可进一步提高多传感器的测量精度和可靠性,适用于多传感器的数据融合.     

Performance limits of track-to-track fusion versus centralized estimation: theory and application [sensor fusion]

Track-to-track fusion is an important part in distributed multisensor-multitarget tracking. The centralized and distributed tracking configurations were studied in (H.Chen et al., Proc. of SPIE Conf. on Signal and Data Processing of Small Targets, vol. 4048, 2000) using simulated air-to-air scenarios, and in (K.C. Chang, et al, IEEE Transact. on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1271-1276, 1997) with analytical results based on /spl alpha/-/spl beta/ filters. The current work generalizes the results in the latter to the cases with more than 2 sensors. As the number of sensors increases, the performance of the distributed tracker is shown to degrade compared with the centralized estimation even when the optimal track-to-track fusion is used. An approximate track-to-track fusion is presented and compared with the optimal track-to-track fusion with performance curves for various numbers of sensors. These performance curves can be used in designing a fusion system where certain trade-offs need to be considered. Finally, these results are compared with simulation results for a realistic air-to-air encounter scenario.     

基于高斯混合模型的航迹抗差关联算法

针对雷达系统误差时变、上报目标不完全一致等复杂场景下目标航迹关联问题,采用高斯混合模型（GMM）与航迹间拓扑信息相结合的方法实现航迹抗差关联。将航迹关联问题转化为图像匹配中的非刚性点集匹配问题,建立对非同源航迹具有鲁棒性的高斯混合模型,根据航迹间的邻域拓扑信息决定高斯混合模型中各高斯组成部分的权重,利用期望最大值（EM）算法求解高斯混合模型的最优闭合解,在期望步（E-step）阶段求解航迹的对应关系,在最大化步（M-step）阶段求解非同源航迹比例,最后进行航迹关联判决以获得关联结果。仿真结果表明,该算法在不同系统误差、目标分布密度、探测概率等环境下具有较好有效性和鲁棒性。     

旋转盘腔稳态换热试验的时间相关性

在稳态换热试验中,试验没有达到稳定就测量会增大误差,常见工况中盘面平均努塞尔数Nu误差的半衰期为2500~2700s。为判断试验是否达到稳定或可测量状态,采用瞬态计算方法对旋转盘腔换热试验的稳定时间的判定方法进行了研究。数值计算结果表明:对于旋转盘腔换热试验,稳定时间较长。提供了一种通过试验精度判定稳定时间和可测时间的方法,试验前可以由固体非稳态导热的傅里叶数估算可测时间。建议试验过程中每隔一段时间观测传感器数值,观测时间间隔按对数规律确定,直到传感器数值不变以确定达到稳定状态。      

Kalman filtering for matrix estimation

A general discrete-time Kalman filter (KF) for state matrix estimation using matrix measurements is presented. The new algorithm evaluates the state matrix estimate and the estimation error covariance matrix in terms of the original system matrices. The proposed algorithm naturally fits systems which are most conveniently described by matrix process and measurement equations. Its formulation uses a compact notation for aiding both intuition and mathematical manipulation. It is a straightforward extension of the classical KF, and includes as special cases other matrix filters that were developed in the past. Beyond the analytical value of the matrix filter, it is shown through various examples arising in engineering problems that this filter can be computationally more efficient than its vectorized version.     

含固支边的强厚度层合板解析解的改进方法

基于含固支边层合板的非齐次状态方程,应用增维方法,建立了齐次状态方程,给出了静力问题的解析解。这种方法对于程序的实现和数值运算稳定性的提高十分有利,在求解的过程中避免了矩阵求逆,且大大提高了计算效率。数值算例显示了本文的方法是正确的。     

The modeling and estimation of asynchronous multirate multisensor dynamic systems

An asynchronous data fusion problem based on a kind of multirate multisensor dynamic system is studied. The system is observed by multirate sensors independently, with the state model known at the finest scale. Under the assumption that the sampling rates of sensors decrease successively by any positive integers, the discrete dynamic system models are established based on each single sensor and an asynchronous multirate multisensor state fusion estimation algorithm is presented. Theoretically, the estimate is proven to be unbiased and the optimal in the sense of linear minimum covariance, the fused estimate is better than the Kalman filtering results based on each single sensor, and the accuracy of the fused estimate will decrease if any of the sensors' information is neglected. The feasibility and effectiveness of the algorithm are shown through simulations.     

非线性不平衡转子轴承系统周期解的预测

本文提出一种对非线性不平衡转子轴承系统周期解进行预测的新型算法,它利用系统周期解的稳态及瞬态信息,反解雅可比矩阵,实现对系统周期解的预测追踪,并利用反解得出的雅可比矩阵,求得系统周期解的Floquet乘子以判别其非线性稳定性。文中以刚性不平衡转子轴承系统为例,实现了周期解的预测追踪及非线性稳定性判别,说明了新算法的有效性。