用正则化多项式建模目标轨迹，平衡精度与简洁性。
From Target Tracking to Targeting Track -- Part II: Regularized Polynomial Trajectory Optimization
- 将轨迹建模为带趋势和残差的随机过程，用多项式拟合时间序列观测。
- 采用阶数限制与ℓ₀正则两种策略，分别通过网格搜索和混合牛顿法优化。
- 在单/多机动目标场景中验证有效，适合需要精确轨迹估计的追踪任务。
目标跟踪旨在估计目标状态随时间的演变，即目标轨迹。不同于经典的状态空间模型，本系列研究（包括本文）将目标状态集合建模为随机过程（SP），进一步分解为表示轨迹趋势的确定性部分和表示残差拟合误差的残差SP。随后，跟踪问题被转化为关于轨迹随机过程的学习问题，关键在于估计最佳拟合时序测量数据的轨迹形式（T-FoT）。为此，本文考虑多项式T-FoT，提出两种正则化策略以在精度与简洁性之间取得平衡：一种限制多项式阶数，通过在窄范围内的网格搜索确定最优阶数；另一种采用ℓ₀范数正则化，并使用混合牛顿求解器进行优化。在单目标和多机动目标场景下的仿真结果证明了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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