Joint Multi-Gaussian Mixture Probability Hypothesis Density Filter for Bearings-only Multi-target Tracking
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摘要: 现有的多模型-高斯混合-概率假设密度(MM-GM-PHD)滤波器被广泛用于不确定机动目标跟踪，但它不能在不同模型下保持并行的估计，导致各模型的似然值滞后于目标机动。为此，该文提出一种联合多高斯混合概率假设密度(JMGM-PHD)滤波器，并将其用于纯方位多目标跟踪。首先，推导了JMGM模型，其中每个单目标状态估计由一组并行的、带模型概率的高斯函数描述，该状态估计的概率由一个非负的权重来表征。一组权值、模型概率、均值和协方差被统称为JMGM分量。根据贝叶斯规则，推导了JMGM分量的更新方法。然后，利用JMGM模型近似多目标PHD。根据交互式多模型(IMM)规则，推导出JMGM分量的交互、预测和估计方法。将所提JMGM-PHD滤波器应用于纯方位跟踪(BOT)时，针对同时执行平移和旋转的观测站，基于复合函数求导规则推导出一种计算线性化观测矩阵的方法。所提JMGM-PHD滤波器保持了单模型PHD滤波器的形式，但能够自适应地跟踪不确定机动目标。仿真结果表明，JMGM-PHD滤波器克服了似然值滞后于目标机动的问题，在跟踪精度和计算成本方面均优于MM-GM-PHD滤波器。Abstract: The Multi-Model Gaussian Mixture-Probability Hypothesis Density (MM-GM-PHD) filter is widely used in uncertain maneuvering target tracking, but it does not maintain parallel estimates under different models, leading to the model-related likelihood lagging behind unknown target maneuvers. To solve this issue, a Joint Multi-Gaussian Mixture PHD (JMGM-PHD) filter is proposed and applied to bearings-only multi-target tracking in this paper. Firstly, a JMGM model is derived, where each single-target state estimate is described by a set of parallel Gaussian functions with model probabilities, and the probability of this state estimate is characterized by a nonegative weight. The weights, model-related probabilities, means and covariances are collectively called JMGM components. According to the Bayesian rule, the updating method of the JMGM components is derived. Then, the multi-target PHD is approximated using the JMGM model. According to the Interactive Multi-Model (IMM) rule, the interacting, prediction and estimation methods of the JMGM components are derived. When addressing Bearings-Only Tracking (BOT), a method based on the derivative rule for composite functions is derived to compute the linearized observation matrix of observers that simultaneously performs translations and rotations. The proposed JMGM-PHD filter preserves the form of regular single-model PHD filter but can adaptively track uncertain maneuvering targets. Simulations show that our algorithm overcomes the likelihood lag issue and outperforms the MM-GM-PHD filter in terms of tracking accuracy and computation cost.
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1 所提JMGM-PHD滤波应用于BOT时的算法
输入：上一时刻PHD $ {v_{k - 1}} $、量测$ {{\boldsymbol{Z}}_k} $、观测站姿态$ (x_k^{\text{O}},y_k^{\text{O}}) $, $ \theta _k^{\text{O}} $ (1) 根据式(12)–式(15)预测PHD，得到式(16)所述的$ {v_{k|k - 1}} $ (2) 根据式(17)–式(21)更新$ {v_{k|k - 1}} $，其中似然的计算见式(29)、
式(30)(3) 剔除权重小于$ {\lambda _{\text{d}}} $的JMGM分量，后根据式(31)–式(35)执行合并 (4) 根据式(24)–式(26)估计目标数、目标状态和多目标模型概率 输出：当前时刻PHD$ {v_k} $，多目标的状态估计$ \hat {\boldsymbol{x}}_k^{(i)} $和模型概率$ u_k^{(m)} $ 
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表 1 各滤波器平均OSPA误差曲线的均值、最大值和标准差
滤波器 均值 最大值 标准差 MM-GM-PHD 50.150 2 97.146 2 10.729 5 MMF-GM-PHD 74.883 5 100.000 0 19.286 3 JMGM-PHD 41.452 3 62.084 0 8.463 9
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