From target tracking to targeting track-Part II: Regularized polynomial trajectory optimization

Target tracking entails the estimation of the evolution of the target state over time, namely the target trajectory. Classical state-space modeling approaches, which focus on estimating discrete-time point states, exhibit notable limitations in accurately representing long-term trajectory trends and in dealing with model mismatch in scenarios with complex maneuvers. Different from the classical state space model, our series of studies, including this paper, model the collection of the target state overtime as a stochastic process (SP) that is further decomposed into a deterministic part which represents the trend of the trajectory and a residual SP representing the residual fitting error. Subsequently, the tracking problem is formulated as a learning task regarding the trajectory SP for which a key part is to estimate a trajectory function of time (T-FoT) best fitting the measurements in time series. For this purpose, we consider the polynomial fitting and address the regularized polynomial T-FoT optimization employing two distinct regularization strategies on the order of the polynomial, seeking trade-off between the accuracy and simplicity. One solves the problem by grid searching in a narrow, bounded range that is proven containing the optimal result while the other adopts ℓ₀ norm regularization for which a hybrid Newton solver is designed. Simulation results obtained in both single and multiple maneuvering target scenarios demonstrate the effectiveness of our approaches.