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From Target Tracking to Targeting Track -- Part II: Regularized Polynomial Trajectory Optimization

Tiancheng Li, Yan Song, Guchong Li, Hao Li

TL;DR

This paper reframes target tracking as learning a continuous-time trajectory through a trajectory function of time (T-FoT) with deterministic-stochastic decomposition, enabling online polynomial fitting of the trend. It introduces two regularization strategies on the polynomial coefficients: order-limiting via a grid-search on $\gamma$ and $\ell_0$-regularization via a hybrid Newton method, along with an efficient Order-Recursive Least Squares (ORLS) solver for linear measurements and nonlinear measurement extensions. Simulation results show that adaptive-order T-FoT fitting substantially outperforms fixed-order and $\ell_1$-regularized approaches in both single and multi-target scenarios, with ORLS offering a promising balance of accuracy and real-time feasibility, while $\ell_0$-Newton achieves the best accuracy at the expense of speed. The work advances continuous-time trajectory estimation with uncertainty–aware learning and provides practical guidance on regularization choices for maneuvering targets. Overall, the proposed framework enhances robustness to maneuvers and irregular data, with strong implications for real-time tracking and future integration with residual SP learning in companion work.

Abstract

Target tracking entails the estimation of the evolution of the target state over time, namely the target trajectory. Different from the classical state space model, our series of studies, including this paper, model the collection of the target state 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 problem regarding the trajectory SP for which a key part is to estimate a trajectory FoT (T-FoT) best fitting the measurements in time series. For this purpose, we consider the polynomial T-FoT and address the regularized polynomial T-FoT optimization employing two distinct regularization strategies seeking trade-off between the accuracy and simplicity. One limits the order of the polynomial and then the best choice is determined by grid searching in a narrow, bounded range while the other adopts $\ell_0$ norm regularization for which the hybrid Newton solver is employed. Simulation results obtained in both single and multiple maneuvering target scenarios demonstrate the effectiveness of our approaches.

From Target Tracking to Targeting Track -- Part II: Regularized Polynomial Trajectory Optimization

TL;DR

This paper reframes target tracking as learning a continuous-time trajectory through a trajectory function of time (T-FoT) with deterministic-stochastic decomposition, enabling online polynomial fitting of the trend. It introduces two regularization strategies on the polynomial coefficients: order-limiting via a grid-search on and -regularization via a hybrid Newton method, along with an efficient Order-Recursive Least Squares (ORLS) solver for linear measurements and nonlinear measurement extensions. Simulation results show that adaptive-order T-FoT fitting substantially outperforms fixed-order and -regularized approaches in both single and multi-target scenarios, with ORLS offering a promising balance of accuracy and real-time feasibility, while -Newton achieves the best accuracy at the expense of speed. The work advances continuous-time trajectory estimation with uncertainty–aware learning and provides practical guidance on regularization choices for maneuvering targets. Overall, the proposed framework enhances robustness to maneuvers and irregular data, with strong implications for real-time tracking and future integration with residual SP learning in companion work.

Abstract

Target tracking entails the estimation of the evolution of the target state over time, namely the target trajectory. Different from the classical state space model, our series of studies, including this paper, model the collection of the target state 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 problem regarding the trajectory SP for which a key part is to estimate a trajectory FoT (T-FoT) best fitting the measurements in time series. For this purpose, we consider the polynomial T-FoT and address the regularized polynomial T-FoT optimization employing two distinct regularization strategies seeking trade-off between the accuracy and simplicity. One limits the order of the polynomial and then the best choice is determined by grid searching in a narrow, bounded range while the other adopts norm regularization for which the hybrid Newton solver is employed. Simulation results obtained in both single and multiple maneuvering target scenarios demonstrate the effectiveness of our approaches.

Paper Structure

This paper contains 20 sections, 5 theorems, 54 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Relevant Work
  3. Introduction to Companion Papers
  4. Contribution and Organization of This Paper
  5. Preliminaries
  6. Polynomial T-FoT
  7. Sliding Time-Window T-FoT Fitting
  8. $\ell_0$-regularization Optimality
  9. Polynomial T-FoT with Limiting Order
  10. Bounds for $\gamma^ *$ and $\lambda$
  11. Bounds of ${\gamma^ * }$
  12. Bounds of $\lambda$
  13. Order-Recursive Least Squares (ORLS) Solver for Linear Measurement
  14. Extension to Nonlinear Measurement
  15. $\ell_0$-regularized T-FoT Fitting
  16. ...and 5 more sections

Key Result

Lemma 1

Let $g$ be an L-smooth function ($L \ge 0$). Then for any $\mathbf{u},\mathbf{w}\in \mathbb{R}^n$, where $\left<\cdot ,\cdot \right>$ represents the inner product.

Figures (10)

  • Figure 1: A taxonomy of existing prominent estimators for target tracking
  • Figure 2: Fitting by polynomials of different orders.
  • Figure 3: Illustration of the monotonous decrease of data fitting error $\mathcal{D}_K(\mathbf{C}_{\gamma})$ and the monotonous increase of order cost with the increase of $\gamma$.
  • Figure 4: Flowchart of the hybrid Newton algorithm for T-FoT $\ell_0$ optimization.
  • Figure 5: Real trajectory and estimates given by different estimators in one trial.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 1: L-Smooth
  • Lemma 1: descent lemma
  • Definition 2: $\ell$-strongly convex
  • Lemma 2
  • Definition 3: $\tau$-stationary
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • ...and 3 more