二维仅测距跟踪中广义von Mises分布的三角矩
Trigonometric Moments of a Generalized von Mises Distribution in 2-D Range-Only Tracking
AI总结：
本文针对二维仅测距跟踪中条件方位密度服从广义von Mises分布且三角矩难以计算的问题，证明了其三角矩可用无穷级数表示并以少量项近似，进一步通过优化问题生成确定性样本实现精确滤波，仿真验证了方法的可行性。
摘要：
二维仅测距跟踪场景具有非平凡性，主要有两个原因。首先，当待估计的状态采用笛卡尔坐标表示时，不确定区域是多模态的。第二个原因是，以距离为条件的方位角概率密度函数呈现广义von Mises分布的形式，这很难处理。即使在实现单模态卡尔曼滤波器的情况下，也需要条件方位密度三角函数的期望，而现有文献中尚无法获得这些期望。我们证明，以距离为条件的方位角密度的三角矩（圆矩）可以计算为无穷级数，并且可以通过求和中相对较少的项进行充分近似。该解还可以推广到任意阶矩。这一重要结果能够准确刻画二维仅测距跟踪几何中条件方位密度的特征。我们还提出了一个简单的优化问题，该问题利用条件方位密度的圆矩知识生成确定性样本，从而实现精确的滤波解。结果在一个二维仿真中展示，其中仅测距传感器平台进行机动以使系统可观测。结果表明该方法在此类应用中具有可行性。
英文摘要：
A 2D range-only tracking scenario is non-trivial due to two main reasons. First, when the states to be estimated are in Cartesian coordinates, the uncertainty region is multi-modal. The second reason is that the probability density function of azimuth conditioned on range takes the form of a generalized von Mises distribution, which is hard to tackle. Even in the case of implementing a uni-modal Kalman filter, one needs expectations of trigonometric functions of conditional bearing density, which are not available in the current literature. We prove that the trigonometric moments (circular moments) of the azimuth density conditioned on range can be computed as an infinite series, which can be sufficiently approximated by relatively few terms in summation. The solution can also be generalized to any order of the moments. This important result can provide an accurate depiction of the conditional azimuth density in 2D range-only tracking geometries. We also present a simple optimization problem that results in deterministic samples of conditional azimuth density from the knowledge of its circular moments leading to an accurate filtering solution. The results are shown in a two-dimensional simulation, where the range-only sensor platform maneuvers to make the system observable. The results prove that the method is feasible in such applications.
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