In this paper, a trajectory shape constraint on target motion is investigated, and the corresponding state estimation method is presented. This type of constraint occurs when only the shape of the target trajectory is known a priori, without any other necessary information to exactly describe the specific trajectory. For example, one may only know that the target moves along a straight line. This prior knowledge of trajectory can be considered as a constraint to improve tracking performance. To describe the shape constraint arising from the straight line assumption, the state vector is augmented by the states at previous time steps. This facilitates the description of the shape constraint using the components of the state vector. Pseudomeasurements are then constructed based on these relationships to incorporate the constraint into an estimation process to improve the performance. The redundancy of the complete set of pseudomeasurements for a given augmented state vector is analyzed, and the minimal set of pseudomeasurements, which describes the constraint exactly, is proposed. The time evolution equation for the augmented state and the measurement equation using the minimal pseudomeasurement set are formulated. A trajectory shape constraint Kalman filter (TSCKF) is then proposed for simultaneous filtering and smoothing. Since both the measurement vector and the state vector are high dimensional, the cubature Kalman filter is used in the proposed TSCKF to deal with the strong nonlinearity in this problem. Monte Carlo simulations illustrate the effectiveness of the proposed TSCKF and the improvement in both filtering and smoothing accuracies by incorporating the trajectory shape constraint.

In transportation networks, vehicle motions are usually subject to the constraints arising from the roads, preset sea-routes, or preset flight routes. Taking advantage of prior known information of such constraints generally produces better surveillance performance. The problem of state estimation with heading constraints, in which only the direction of the target trajectory is prior known, is considered. When only part of the trajectory information is available, the conventional constrained estimation methods cannot be used to produce constrained estimates. The heading constraints in two typical situations are investigated, one is with a straight line, the other is a circular arc. For the straight trajectory, two augmentation approaches are proposed to formulate the constraints. One, parameter augmentation, augments the base state by the $y$ -intercept of the constraint straight line. In the other approach, state augmentation, the states at the past time step are used to augment the base state. Two corresponding heading constraints are formulated by the augmented state’s elements and the prior known information about road direction. For a circular trajectory, the heading constraint is formulated using position and velocity components in the base state. Pseudo-measurements are constructed to incorporate the heading constraint into the estimators and result in three heading constraint Kalman filters (HCKFs). They are parameter augmentation HCKF and state augmentation HCKF in case of straight trajectory, and HCKF for circular trajectory. The discrimination of road segments at junctions is also discussed. Furthermore, the proposed constraint filters are integrated into the interacting multiple model estimator to handle on-road maneuvers with possible accelerations. Numerical simulations are conducted to evaluate the performances of the three HCKFs compared with existing methods.

In some target tracking scenarios, tracking performance can be improved by the incorporation of a constraint imposed by a preset trajectory. When the complete information about the trajectory is known, the constraint can be formulated in an explicit way. However, in practical applications, only partial information about the trajectory may be available. This paper deals with the modeling of implicit constraints, imposed by a circular trajectory, along with the constrained state estimation. First, constraint models for two kinds of representative implicit constraints, the destination constraint and the trajectory shape constraint, are proposed for various cases with different conditions of prior information. In the cases with destination constraints, the destination information and the trajectory shape information are assumed to be known a priori. For each case, two forms of constraint models are derived. One is a direct form model where the constraint relationships are used to eliminate the unknowns in the final model. While the other is a state augmentation form model, where the unknowns are augmented into the state vector being estimated along with the target state. In the cases with trajectory shape constraints, only the trajectory shape information is available. For each cases, the state augmentation form models are formulated. Second, constrained state estimation methods are proposed by utilizing pseudo-measurement constructed based on the constraint models. Furthermore, an extension to the maneuvering target tracking by integrating the proposed constraint filters into an interacting multiple model estimator is also presented. Numerical simulations in two scenarios are provided to show the effectiveness of the proposed constraint models and state estimation methods.

In some target tracking applications, angle measurements are absent due to sensor characteristics, which makes Cartesian states unobservable for a single stationary sensor. When multi-sensor target tracking techniques with only range or Doppler measurements are applied to produce Cartesian state estimates, their performance and efficiency can be benefited from accurate range and Doppler estimates obtained from single sensors. This paper investigates the range and Doppler estimation problem based on motion modeling in range-squared coordinate considering range-squared as a generic coordinate. A pseudostate vector consisting of range squared and its derivatives is defined to completely summarize the dynamic of the Cartesian nearly constant velocity (NCV) motion and it is found time evolution of the pseudostate is linear based on the methodology of state space representation. To estimate the pseudostate from range-Doppler or range-only measurements, a recursive filter is developed using the decorrelated unbiased converted measurement (DUCM) method. Finally, the pseudostate estimates in the range-squared coordinate are converted into the range coordinate using the scaled unscented transformation (SUT) method to produce range and Doppler estimates. Although the conversion is nonlinear, it is performed outside the filtering recursions without propagating nonlinearity approximation errors in the filtering recursions. Numerical experiments with various conditions of measurements demonstrate the superior performance benefits from the proposed accurate linear state equation and estimation method.