Ting Yuan

A Multiple IMM Estimation Approach with Unbiased Mixing for Thrusting Projectiles

We present a procedure to estimate the state of thrusting/ballistic endoatmospheric projectiles for the end purpose of impact point prediction. The short observation time and the estimation ambiguity between drag and thrust in the dynamic model motivate the development of a multiple interacting multiple model (MIMM) estimator with various drag coefficient initializations. A simple unbiased IMM mixing procedure (useful for quite general applications) is presented for state estimators with unequal dimensions and applied for the thrusting and ballistic modes in the case considered. Results with real data are given.

Impact Point Prediction for Thrusting Projectiles in the Presence of Wind

Wind can and often does significantly affect impact-point prediction (IPP) performance for thrusting/ballistic endoatmospheric projectiles. Wind exacerbates the estimation ambiguity between drag and thrust in the dynamic model and induces additional uncertainty in the IPP procedure. A tracker accounting for the wind effect is presented and simulation study shows that it can be fully compensated if the wind information is available. An N-point adaptive initialization based on a goodness-of-fit test and a statistical significance test is introduced. Based on the multiple interacting multiple model (MIMM) approach developed recently, the IPP performance is investigated with respect to the total observation time and the sensor accuracy in various wind scenarios. In each Monte Carlo (MC) run of the simulation study, under the same sensor accuracy and the same observation time, the same set of random numbers has been used (but different in different MC runs) for the same caliber projectile in various wind scenarios to examine how much the wind affects the IPP performance with/without the exact knowledge of the wind information. The final conclusion is that with the wind effect accounted for, the IPP performance in the presence of wind is practically the same as in its absence.

Impact point prediction for short range thrusting projectiles

This paper presents an interacting multiple model (IMM) based procedure to estimate the state of thrusting ballistic projectiles in the atmosphere for the purpose of impact point prediction (IPP). The modes of the IMM estimator are for the thrusting and the ballistic phases and different extended Kalman filters (EKF) are used as the mode-matched filters with different dimension states. The IPP is achieved by using the IMM-predicted most probable mode at the mid-point of the trajectory.

Estimation of thrusting trajectories in 3D from a single fixed passive sensor

The problem of estimating the state of thrusting/ballistic endo-atmospheric projectiles moving in three-dimensional space for the purpose of impact point prediction (IPP) using two-dimensional measurements from a single passive sensor (stationary or moving with constant velocity) is investigated. The location of a projectiles launch point (LP) is generally unavailable, and this could significantly affect the performance of the estimation and the IPP. However, if the altitude of the LP is known, the launch position can be obtained with negligible error from the first line of sight measurement intersected with the terrain map. The estimability is analyzed based on the Fisher Information Matrix (FIM) of the target parameter vector that determines its trajectory: the initial launch (azimuth and elevation) angles, drag coefficient, and thrust. Lack of knowledge about the LP altitude makes the problem substantially more difficult, since this altitude is then an additional unknown target parameter and must be included into the target parameter vector that needs estimability analysis. The full rank of the FIM, with/without the LP altitude, ensures that one has estimable target parameters. The corresponding Craḿer-Rao lower bound quantifies the estimation performance of the estimator that is statistically efficient and can be used for the IPP accuracy evaluation. In view of the inherent nonlinearity of the problem, the maximum likelihood estimate of the target parameter vector can be found by using a suitable numerical approach. A search strategy with two stages-a mixed (partially grid-based) search followed by a continuous search-is proposed. For even a coarse grid, this approach is shown to have reliable estimation performance and leads to an IPP of good accuracy. Due to its parallelizable nature, the mixed search allows the two-stage strategy to be implementable in real time.

A Generalized Information Matrix Fusion Based Heterogeneous Track-to-Track Fusion Algorithm ∗

The problem of Track-to-Track Fusion (T2TF) is very important for distributed tracking systems. It allows the use of the hierarchical fusion structure, where local tracks are sent to the fusion center (FC) as summaries of local information about the states of the targets, and fused to get the global track estimates. Compared to the centralized measurement-to-track fusion (CTF), the T2TF approach has low communication cost and is more suitable for practical implementation. Although having been widely investigated in the literature, most T2TF algorithms dealt with the fusion of homogenous tracks that have the same state of the target. However, in general, local trackers may use different motion models for the same target, and have different state spaces. This raises the problem of Heterogeneous Track-to-Track Fusion (HT2TF). In this paper, we propose the algorithm for HT2TF based on the generalized Information Matrix Fusion (GIMF) to handle the fusion of heterogenous tracks in the presence of possible communication delays. Compared to the fusion based on the LMMSE criterion, the proposed algorithm does not require the crosscovariance between the tracks for the fusion, which greatly simplify its implementation. Simulation results show that the proposed HT2TF algorithm has good consistency and fusion accuracy.

A multiple IMM approach with unbiased mixing for thrusting projectiles

This paper presents a multiple interacting multiple model (MIMM) procedure to estimate the state of thrusting/ ballistic projectiles in the atmosphere for the purpose of impact point prediction (IPP). Given a very short time span of observations, the strong interaction between drag and thrust in the dynamic model, in the sense of ambiguity in the estimation, significantly affects the estimation performance and the final IPP accuracy. This leads to the need to use an MIMM estimator with various initial drag coefficient estimates. The modes of each IMM estimator are for the thrusting and the ballistic phases and different extended Kalman filters (EKF) are used as the mode-matched filters with different dimension states. A novel unbiased mixing procedure for an IMM estimator is introduced to deal with state estimates with unequal dimensions, as is the case for the thrusting and ballistic models. The IPP is carried out at the end of the observation period by using the most probable mode of the selected IMM estimator, the latter being the one with the highest likelihood in the MIMM approach.

Estimability of thrusting trajectories in 3-D from a single passive sensor with unknown launch point

The problem of estimating the state of thrusting/ballistic endoatmospheric projectiles moving in 3-dimensional (3-D) space using 2-dimensional (2-D) measurements from a single passive sensor is investigated. The location of projectile’s launch point (LP) is unavailable and this could significantly affect the performance of the estimation and the IPP. The LP altitude is then an unknown target parameter. The estimability is analyzed based on the Fisher Information Matrix (FIM) of the target parameter vector, comprising the initial launch (azimuth and elevation) angles, drag coefficient, thrust and the LP altitude, which determine the trajectory according to a nonlinear motion equation. The full rank of the FIM ensures that one has an estimable target parameters. The corresponding Cram´er-Rao lower bound (CRLB) quantifies the estimation performance of the estimator that is statistically efficient and can be used for IPP. In view of the inherent nonlinearity of the problem, the maximum likelihood (ML) estimate of the target parameter vector is found by using a mixed (partially grid-based) search approach. For a selected grid in the drag-coefficient-thrust-altitude subspace, the proposed parallelizable approach is shown to have reliable estimation performance and further leads to the final IPP of high accuracy.

Estimability of thrusting trajectories in 3D from a single passive sensor

The problem of estimating the state of thrusting/ballistic endoatmospheric projectiles moving in 3-dimensional (3-D) space using 2-dimensional (2-D) measurements from a single passive sensor (stationary or moving with constant velocity) is investigated. The estimability is analyzed based on the Fisher Information Matrix (FIM) of the target parameter vector, comprising the initial launch (azimuth and elevation) angles, drag coefficient and thrust, which determine its trajectory according to a nonlinear motion equation. The initial position is assumed to be obtained from the first line of sight (LoS) measurements intersected with a known-altitude plane. The full-rank FIM ensures that this is an estimable system. The corresponding Cram´er-Rao lower bound (CRLB) quantifies the estimation performance of the estimator that is statistically efficient and can be used for impact point prediction (IPP). Due to the inherent nonlinearity of the problem, the maximum likelihood estimate of the target parameter vector is found by using iterated least squares (ILS) numerical approach. A combined grid and ILS approach searches over the launch angles space is proposed. The drag coefficient-thrust grid-based ILS approach is shown to converge to the global maximum and has reliable estimation performance. This is then used for IPP.

Track-to-Track Fusion in Linear and Nonlinear Systems

This chapter starts with a review of the architectures for track-to-track fusion (T2TF). Based on whether the fusion algorithm uses the track estimates from the previous fusion and the configuration of information feedback, T2TF is categorized into six configurations, namely, T2TF with no memory with no, partial and full information feedback, and T2TF with memory with no, partial and full information feedback. The exact algorithms of the above T2TF configurations and the impact of information feedback on fusion accuracy are presented. Although (under the Linear Gaussian assumption) the exact T2TF algorithms yield theoretically consistent fusion results, their major drawback is the need of the crosscovariances of the tracks to be fused, which drastically complicates their implementation. The information matrix fusion (IMF) is a special case of T2TF with memory. Although it is heuristic when not conducted at full rate, it was shown to have consistent and near optimal fusion performance for practical tracking scenarios. Due to its simplicity, it is a good candidate for practical tracking systems. For the problem of asynchronous T2TF (AT2TF), a generalized version of the IMF is presented. It supports information feedback for AT2TF in the presence of communication delay, and was shown to have good consistency and close to optimal fusion accuracy. Finally the fusion of heterogenous tracks where the states at the local trackers are nonlinearly related and of different dimension is discussed. For the problem of the fusion of the track from an Interacting Multiple Model (IMM) estimator from an active sensor with the track from a passive sensor, a counterintuitive phenomenon that heterogenous T2TF may have better performance than the centralized measurement-to-track fusion approach (which is the known optimum in the linear case) is demonstrated and explained.

Impact point prediction for thrusting projectiles in the presence of wind

In estimating the state of thrusting/ballistic endoatmospheric projectiles for the end purpose of impact point prediction (IPP), the total observation time, the wind effect and the sensor accuracy significantly affect the IPP performance. First the tracker accounting for the wind effect is presented. Following this, based on the multiple interacting multiple model (MIMM) estimator developed recently, a sensitivity study of the IPP performance with respect to the total observation time, the wind (strength and direction) and the sensor accuracy is presented.