Ronen Ben-Dov

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.

Passive ranging using signal intensity observations from a single fixed sensor

A passive ranging problem with elevation angle, azimuth angle and signal intensity measurements is presented and solved with a Maximum Likelihood (ML) estimator. The measurements used in the estimation are all obtained from a single passive sensor at a fixed location. The intensity measurement, which obeys the inverse square law w.r.t. the squared distance between the sensor and target has an unknown emitted energy that needs to be taken into account in the estimation problem. The Fisher Information Matrix (FIM) is investigated and used for observability testing. The simulation results from the scenarios considered prove the efficiency of the ML estimator.

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.

Statistically Efficient Passive Ranging Using Signal Intensity Observations From a Single Fixed Sensor

In this paper, a passive ranging problem with elevation angle, azimuth angle, and signal intensity measurements is presented. The signal intensity is inversely proportional to the squared distance between target and sensor. All measurements are obtained from a single fixed sensor and the emitted signal intensity is unknown. The observability is investigated through the Fisher information matrix (FIM) and Cramer–Rao lower bound. The maximum-likelihood (ML) estimator is used for the estimation of the trajectory of the target and the FIM is calculated numerically. The simulation results prove the feasibility of passive ranging using signal intensity measurements from a single stationary passive sensor. The estimates are shown to be statistically efficient.

Trajectory estimation and impact point prediction of a ballistic object from a single fixed passive sensor

For a thrusting/ballistic target, works have shown that a single fixed sensor with 2-D angle-only measurements (azimuth and elevation angles) is able to estimate the target’s 3-D trajectory. In previous works, the measure- ments have been considered as starting either from the launch point or with a delayed acquisition. In the latter case, the target is in flight and thrusting. The present work solves the estimation problem of a target with delayed acquisition after burn-out time (BoT), i.e. in the ballistic stage. This is done with a 7-D parameter vector (velocity vector azimuth angle and elevation angle, drag coefficient, 3-D acquisition position and target speed at the acquisition time) assuming noiseless motion. The Fisher Information Matrix (FIM) is evaluated to prove the observability numerically. The Maximum Likelihood (ML) estimator is used for the motion parameter estimation at acquisition time. The impact point prediction (IPP) is then carried out with the ML estimate. Simulation results from the scenarios considered illustrate that the MLE is efficient.

Tracking Initially Unresolved Thrusting Objects Using an Optical Sensor

This paper considers the problem of estimating the three-dimensional states of a salvo of thrusting/ballistic endo-atmospheric objects using two-dimensional (2-D) Cartesian measurements from the focal plane array (FPA) of a single fixed optical sensor. Since the initial separations in the FP are smaller than the resolution of the sensor, there are merged FP measurements, compounding the usual false-alarm and missed-detection uncertainty. We present a two-step methodology. First, we assume a Wiener process acceleration model for the motion of the images of the objects in the optical sensors FPA. We model the merged measurements with increased variance, and thence employ a multi-Bernoulli (MB) filter using the 2-D measurements in the FPA. Second, using the set of associated measurements for each confirmed MB track, we formulate a parameter estimation problem, whose maximum likelihood solution can be obtained via numerical search and can be used for impact point prediction. Simulation results illustrate the performance of the proposed method.

Motion parameter estimation of a thrusting/ballistic object from a single fixed passive sensor with delayed acquisition

In previous works, it has been shown that the estimation problem of a thrusting/ballistic object in the three-dimensional space can be solved with two-dimensional measurements (azimuth and elevation angles starting from the launch time) assuming the launch point is perfectly known. In this paper, the problem is extended to estimate the targets trajectory with measurements starting after the launch time, i.e., delayed acquisition. Compared to the situation of acquisition at launch time, one has an additional unknown speed (magnitude of the velocity vector) and the unknown acquisition location. The 2D angle measurements are all obtained from a single fixed passive sensor. The parameter vector, in this case, has dimension 8 (velocity vector azimuth angle and elevation angle, drag coefficient, specific thrust, target speed and 3D acquisition position). The invertibility of the Fisher Information Matrix (FIM) of the parameter vector is investigated to test the observability (estimability) of the system. The simulation results prove the statistical efficiency and unbiasedness of the Maximum Likelihood estimator, that is, the Cramer-Rao lower bound (the inverse of the FIM if it is invertible) can be used as the actual covariance.

Tracking initially unresolved thrusting objects in 3D using a single stationary optical sensor

This paper considers the problem of estimating the 3D states of a salvo of thrusting/ballistic endo-atmospheric objects using 2D Cartesian measurements from the focal plane array (FPA) of a single fixed optical sensor. Since the initial separations in the FPA are smaller than the resolution of the sensor, this results in merged measurements in the FPA, compounding the usual false-alarm and missed-detection uncertainty. We present a two-step methodology. First, we assume a Wiener process acceleration (WPA) model for the motion of the images of the projectiles in the optical sensor’s FPA. We model the merged measurements with increased variance, and thence employ a multi-Bernoulli (MB) filter using the 2D measurements in the FPA. Second, using the set of associated measurements for each confirmed MB track, we formulate a parameter estimation problem, whose maximum likelihood estimate can be obtained via numerical search and can be used for impact point prediction. Simulation results illustrate the performance of the proposed method.

A comparison of multiple-IMM estimation approaches using EKF, UKF, and PF for impact point prediction

We discuss a procedure to estimate the state of thrusting/ballistic endoatmospheric projectiles for the purpose of impact point prediction (IPP). 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. In each IMM estimator used, as the mode-matched state estimators for its thrusting mode and ballistics mode are of unequal dimension, an unbiased mixing is required. We explore the MIMM estimator with unbiased mixing (UM) using extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF). For 30 real trajectories, the IPP based on the MIMM-UM estimation approach is carried out with various sets of tuning parameters selected. The MIMM-UM-EKF, MIMM-UM-UKF and MIMM-UM-PF are compared based on the resulting IPP performance, estimator consistency and computational complexity.