This paper is the second part in a series that provides a comprehensive survey of maneuvering target tracking without addressing the so-called measurement-origin uncertainty. It surveys motion models of ballistic targets used for target tracking. Models for all three phases (i.e., boost, coast, and reentry) of motion are covered.

Survey of Maneuvering Target Tracking. Part II: Motion Models of Ballistic and Space Targets

In multitarget scenarios, kinematic constraints from the interaction of targets with their environment or other targets can restrict target motion. Such motion constraint information could improve tracking performance if effectively used by the tracker. In this paper, we propose three particle filtering methods that incorporate constraint information in their proposal and weighting process; the number of targets is fixed and known in all methods. The reproposed constrained motion proposal (RCOMP) utilizes an accept/reject method to propose particles that meet the constraints. The truncated constraint motion proposal (TCOMP) uses proposal densities truncated to satisfy the constraints. The constraint likelihood independent partitions (CLIP) method simply rejects proposed partitions that do not meet the constraints. We use simulation to evaluate the performance of these three methods for two constrained motion scenarios: a vehicle convoy and soldiers executing a leapfrog motion. Moreover, we demonstrate the utility of constraint information by comparing the proposed algorithms with the independent partition (IP) proposal method that does not use constraint information. The simulation results demonstrate that the root mean square error (RMSE) tracking performance of the RCOMP and the TCOMP methods is much better than the CLIP and IP methods; this is due to their more efficient proposal process.

Sequential Monte Carlo Methods for Tracking Multiple Targets With Deterministic and Stochastic Constraints

Compressive sensing and processing handles high-resolution delay-Doppler radar measurements using a low sampling rate, simple receiver design, and inexpensive processing when Nyquist rate sensing and processing becomes impractical. The benefits of compressive sensing and processing are, however, offset by an increase in estimation error that is introduced when processing compressed measurements versus measurements sampled at the Nyquist rate. In this work, an adaptive compressive sensing and processing method is proposed for the radar tracking problem. The adaptive scheme naturally incorporates information on target state that is readily available from a particle filter based tracker. The proposed method is shown to improve tracking performance over nonadaptive compressive sensing and processing, while maintaining low-sampling rates, a computationally inexpensive operation, and a simple receiver design.

Adaptive Compressive Sensing and Processing of Delay-Doppler Radar Waveforms

A reduced state estimator is derived for systems with bounded parameters as inputs. Optimal filter gains are derived for minimizing the total covariance of the estimation error due to measurement noise and parameter uncertainty. It is shown that these filter gains for a two-state system with a Gaussian parameter satisfy the Kalata relation in steady state. Equations are also derived for optimally filtering measurements in arbitrary time order. This reduced state estimator offers novelties over a traditional Kalman filter in its application to the class of problems considered. The total error covariance, which is minimized, makes no use of plant noise. Furthermore, the filter is easier to optimize in high dimensional and multiple sensor applications as well as in processing out-of-sequence measurements.

Reduced state estimator for systems with parametric inputs

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.

Reduced state estimators for consistent tracking of maneuvering targets

For tracking systems with a uniform data rate and stationary measurement noise, non-manoeuvring targets can be accurately tracked with a steady-state Kalman filter. The steady-state Kalman filter, which can be viewed as equivalent to an alpha-beta filter, has been widely applied to many different systems. A means of selecting the filter coefficients was proposed by Kalata (1984) using Kalman filter considerations. An alternative method based on noise reduction ratios is presented in this paper. Using a design criteria with the Kalata relation, optimal filter coefficients can be selected for specific applications. This method generalizes current methods for selecting the filter coefficients.

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Filter coefficient selection using design criteria

With the shift in focus of the missile defense program from intercept of a ballistic missile during reentry to intercept during ascent, the tracking of ballistic missiles under boost is more important. The paper compares filtering performance using standard approaches to that using compensation from acceleration profiles. The results indicate the acceleration compensation is beneficial when applied to either constant velocity or constant acceleration track filters.

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Track filtering of boosting targets

The general solution of the equations that arise in constant coefficient filter theory are solved for arbitrary number of dimensions. Namely, we solve the first order matrix equation for the special case of scalar measurements. We also determine the covariance matrix for the general case. We then illustrate these results for the specific case of an alpha-beta filter.

G.J. Foster

Papers

Filter coefficient selection using design criteria

Track filtering of boosting targets

Solutions of matrix equations arising in track filter theory