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Positioning and tracking based applications are playing an increasing important role in the lives of all citizens and have also been a long standing research topic. In this letter, we consider a specific type of target that moves on a deterministic, predefined trajectory which can be modeled as a curve in the planar space, such as a train/subway that moves on the railway, a car on the highway, a satellite on its orbit, etc. Apart from the a-priori “trajectory” geometry, there is no further (statistical) information about the target motion, e.g, regarding its velocity, acceleration, etc. This makes it intractable to model the target motion by simply assuming a (constant) velocity, acceleration or turn rate. Moreover, the real target trajectory may be deterministic (while being time-varying and unknown) rather than random, which is not readily cast to applying an off-the-shelf Bayesian filter that typically assumes the state and observation sequences as random quantities. In fact in this case, the Bayesian Cram´er-Rao lower bounds do not provide a lower bound for the conditional mean square error (MSE) matrix .
To solve such a trajectory-constrained positioning problem, we investigate both statistical state space model (SSM) for Bayesian filtering and smoothing, and least squares fitting (LSF) framework.
The only random quantity to be inferred is the movement of the target on the trajectory curve, which we call the Movement on Curve (MoC). With respect to the MoC, we establish a trajectory-constrained state space model to implement the unscented filtering and a constrained MoC-trajectory function of time for least squares fitting. A maneuvering target that moves on a simple deterministic trajectory is studied. The results demonstrate a significant performance improvement gained by using the trajectory constraint.
Attached is an illustrative example; explanation is given in the following Letter
T. Li, Single-Road-Constrained Positioning Based on Deterministic Trajectory Geometry, IEEE Communications Letters, vol.23, no.1, 2019, pp. 80-83. IEEEXplore
Target Tracking with Equality/Inequality Constraints Based on Trajectory Function of Time
Jinyang Zhou, Tiancheng Li, Xiaoxu Wang, Litao Zheng
Abstract—This letter addresses the constrained target tracking problem based on the approach of the trajectory function of time (T-FoT). Both state equality and inequality constraints such as the trajectory geometry and width are considered, respectively. The penalty function is used in the T-FoT framework to account for the constraint. More specifically, the logarithmic barrier function is used to solve the challenging inequality constraint. Different from the existing works which are mostly based on constrained Bayesian filters, our approaches make full use of constraints with little approximation. Three representative scenarios have been considered in the simulation for demonstrating the performance of our proposed approaches in comparison with existing constrained approaches based on Bayes filters.
• J. Zhou, T. Li*, X. Wang and L. Zheng, Target Tracking with Equality/Inequality Constraints Based on Trajectory Function of Time, in IEEE Signal Processing Letters, vol. 28, pp. 1330-1334, 2021 @IEEE Xplore
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