Gongjian Zhou

Statically Fused Converted Position and Doppler Measurement Kalman Filters

The work presented in this paper makes two contributions for exploiting Doppler (range rate) measurements in tracking systems. First, a new linear filter, the converted Doppler measurement Kalman filter (CDMKF), is presented to extract nonlinear pseudostates from converted Doppler measurements (i.e., the product of the range measurements and Doppler measurements). The pseudostates are constructed from the converted Doppler and its derivatives. The linearly evolving equations of the pseudostates are derived for common target motion models. The second contribution of this paper is using the CDMKF along with the converted position measurement Kalman filter (CPMKF), in which only the position measurements are used, to establish a new filtering structure, statically fused converted measurement Kalman filters (SF-CMKF). The resulting states of CPMKF and CDMKF are combined by a static minimum mean squared error (MMSE) estimator, where the nonlinearity and correlation between the pseudostates and the Cartesian states are handled simultaneously, to yield the final state estimates. The dynamic nonlinear estimation problem is converted into dynamic linear estimation followed by static nonlinear fusion. The estimation accuracy can be enhanced by incorporating the Doppler measurements via the linear CDMKF, while the filtering stability can be improved by dealing with nonlinearity outside the filtering recursions. Monte Carlo simulations and comparison with the posterior Cramer-Rao bound demonstrate the effectiveness of the CDMKF and SF-CMKF.

Pseudo states estimation for maneuvering target tracking in Doppler radar systems

Maneuvering target tracking with Doppler measurements is investigated. This paper presents a linear filtering method, converted Doppler measurements Kalman filter (CDMKF), to estimate a pseudo state vector, which consists of the converted Doppler and its derivatives. Then the output of CDMKF is combined with the output of converted position measurements Kalman filter (CPMKF) by a nonlinear static estimator outside the filtering recursions. This constructs a new state estimator which is named as parallel converted measurements Kalman filter (PRL-CMKF). Simulation results demonstrate the effectiveness and robustness of the proposed CDMKF and PRL-CMKF.

State estimation with a destination constraint using pseudo-measurements ☆

Abstract The problem of state estimation with a destination constraint, in which only one point on a straight line is known, is considered. The existing estimation methods with linear equality constraint cannot be applied to produce constrained estimates due to the lack of full information of the straight line. Here, two pseudo-measurements are constructed to formulate the destination constraint. One, a noisy pseudo-measurement, uses the measured position as another point on the constraint line. In that case, the implicit constraint on the velocity components is approximated. The other, a noiseless pseudo-measurement, describes directly the deterministic relationship among position and velocity components, based on the constraint. These two measurements are augmented into the measurement vector and result in two different destination constraint Kalman filters (DCKF), both of which utilize the unscented Kalman filter (UKF) to handle measurement nonlinearities. In the DCKF with noisy pseudo-measurement, the unscented transform (UT) is applied to obtain the statistical properties of the augmented measurements before utilizing the UKF. In the DCKF with noiseless pseudo-measurement, both block and sequential processing methods are presented. Additionally, the constraint error is defined as a new measure to evaluate estimation performances with equality constraints. Simulations illustrate the effectiveness of the proposed methods.

Statically Fused Converted Measurement Kalman Filters for Phased-Array Radars

In this paper, statically fused converted measurement Kalman filters (SF-CMKF) are developed for target tracking using measurements reported by phased array radars in direction cosine coordinates. First, the conversions of position and Doppler measurements and the estimation of the mean and variance of the converted measurement errors are explicitly derived. Then, the filtering procedure of the SF-CMKF working in Direction Cosine coordinates (SF-CMKFcos) is formulated. The pseudostate vector is constructed and the pseudostate equation for the nearly constant velocity motion model in three-dimensional Cartesian coordinates is deduced. The converted Doppler measurement Kalman filter (CDMKF) and converted position measurement Kalman filter (CPMKF) are developed to extract information from position and Doppler measurements in Direction Cosine coordinates, respectively. To generate the final target state estimates, the pseudostate estimates from the CDMKF and the Cartesian-state estimates from the CPMKF are fused statically under the minimum mean squared error criterion. The nonlinear static fusion procedure is maintained outside the dynamic filtering recursions, which keeps the nonlinear approximation errors from being accumulated recursively. Finally, a comprehensive performance comparison is carried out using numerical simulations, where the proposed SF-CMKF is evaluated against several commonly used filters that incorporate Doppler measurements for tracking in Direction Cosine coordinates. Simulation results indicate that the proposed filter is superior to the existing filters, especially in extreme situations where the position measurement errors are large.

Fixed-lag smoothing with linear equality constraints

The problem of fixed-lag smoothing with linear equality constraint (LEC) is considered. Fixed-lag smoothing algorithms are developed by applying the state-augmentation approach to several popular constrained filtering methods, including model reduction, pseudo measurement, estimate projection and other two LEC filtering methods based on conversion or direct elimination method. After a briefly review, they are extended to address the smoothing problem with LEC. The nonlinear radar measurements in polar coordinate are converted to Cartesian coordinate by applying unbiased measurement conversion method. Thus the nonlinear smoothing problem is converted to a linear one and the proposed smoothing algorithms are formulated for linear system. The performance of the proposed smoothers are evaluated using a comparison against traditional unconstrained fixed-lag smoothing algorithm. Monte-Carlo simulation results are presented to illustrate the effectiveness of fixed-lag smoothing algorithms with LEC, where a significant performance improvement over the unconstrained smoothing algorithm can be observed. And the impact of the lag (time delay) on the performance is discussed. A small lag is recommended in the practical applications.

Maneuvering target tracking in constraint coordinates with radar measurements

Maneuvering target tracking in constraint coordinates have attracted much research attention in recent years. This paper follows the one-dimension (1D) motion modeling method and focuses on target tracking in constrained road coordinates. An improved initialization method which combines the measurements in different dimensions is presented for the constraint coordinates Kalman filter (CCKF). The CCKF is evaluated with a comprehensive comparison to the state-of-art linear equality constraint estimation methods. Original radar measurements are transformed from polar coordinate to Cartesian coordinate by debiased converted measurement (DCM) method. Numerical simulation results demonstrate the better performance of the CCKF. The filtering consistency of the CCKF is proved by normalized estimation error squared (NEES) test. Then, the interacting multiple model CCKF (IMM-CCKF) is proposed to illustrate the advantages of the CCKF in maneuvering target tracking with spatial equality constraints. The effectiveness of the IMM-CCKF is demonstrated by numerical experiments.

Non-parametric multi-target tracking with Doppler measurements

This paper investigates the problem of incorporating Doppler measurements to improve multi-target tracking performance. The clutter density in both spatial domain and Doppler direction is assumed to be unknown and non-homogeneous. Instead of modeling the clutter density as the product of the clutter spatial density and the clutter Doppler probability density function (pdf), this paper introduces a hyper-spatial density concept, where the Doppler direction is considered as an additional pseudo position dimension. The spatial sparsity estimator is extended to the hyper-spatial density estimation and is combined with the Joint Integrated Probabilistic Data Association (JIPDA) tracker to deal with non-parametric multi-target tracking problem with Doppler measurements. Monte Carlo simulation results demonstrate the effectiveness of the proposed approach.

Sub-sequence principal component approximation denoising for radar high resolution range profiling

Radar high resolution range profiling is an effective means for target feature analysis and target recognition. In practice, radar return is usually contaminated by strong noise and clutter. This results in profile distortion and recognition performance degradation. To improve the quality of profile, in this paper, we present a new denoising method to improve the signal-to-noise ratio (SNR) of the return. The return is segmented into sub-sequences by sliding-window to construct sub-sequence matrix. We demonstrate that the signal components and noise part can be separated in subspace. The SNR improved return is recovered by the sub-sequence principal components approximate. Experimental results show that this approach can effectively enhance the SNR, leading to a high-quality profile.

Constant turn model for converted Doppler measurement Kalman filter

In this paper, we derive the discrete temporal evolution equation of the pseudo state vector, defined by the converted Doppler (the productive of target true range and range rate) and its first derivative, for the constant turn (CT) motion. The resulted linear state equation allows using of linear Kalman filter to extract information from the pseudo measurements (the productive of range and Doppler measurements) of a target moves with constant speed and constant turn rate. The method is referred to as converted Doppler measurement Kalman filter (CDMKF) and is used in parallel with the converted position measurement Kalman filter (CPMKF) to establish a parallel filtering structure (PRL-CMKF). The validity of the proposed CT model is demonstrated by assessing the performance of the CDMKF and PRL-CMKF. Comparative results show the superior performance of the proposed method especially in challenging scenario with large position measurement errors.

Analysis of two-dimensional spectrum for geosyncheronous SAR

Geosynchronous orbital synthetic aperture radar (GEOSAR) can provide the continuous imaging on the larger coverage area in the short revisit period of nearly 24h. However, the difficulties in the GEOSAR imaging process are curve trajectory and the delay effects mainly stem from the path delay and the atmospheric delay. The curve track problem has been perfectly solved by using the method of Taylor expansion. Nevertheless, the total delay is often ignored, causing the incoherent superposition of GEOSAR echoes and the defocusing imaging results. In this paper, the two-dimensional point target spectrum is deduced based on the above all GEOSAR properties. And the validity of the spectrum is validated well by the simulation of the point target imaging.