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Estimation of thrusting trajectories in 3D from a single fixed passive sensor

机译：通过单个固定式无源传感器估算3D中的推力轨迹

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摘要

The problem of estimating the state of thrusting/ballistic endo-atmospheric projectiles moving in three-dimensional space for the purpose of impact point prediction (IPP) using two-dimensional measurements from a single passive sensor (stationary or moving with constant velocity) is investigated. The location of a projectile???s launch point (LP) is generally unavailable, and this could significantly affect the performance of the estimation and the IPP. However, if the altitude of the LP is known, the launch position can be obtained with negligible error from the first line of sight measurement intersected with the terrain map. The estimability is analyzed based on the Fisher Information Matrix (FIM) of the target parameter vector that determines its trajectory: the initial launch (azimuth and elevation) angles, drag coefficient, and thrust. Lack of knowledge about the LP altitude makes the problem substantially more difficult, since this altitude is then an additional unknown target parameter and must be included into the target parameter vector that needs estimability analysis. The full rank of the FIM, with/without the LP altitude, ensures that one has estimable target parameters. The corresponding Cram??er???Rao lower bound quantifies the estimation performance of the estimator that is statistically efficient and can be used for the IPP accuracy evaluation. In view of the inherent nonlinearity of the problem, the maximum likelihood estimate of the target parameter vector can be found by using a suitable numerical approach. A search strategy with two stages???a mixed (partially grid-based) search followed by a continuous search???is proposed. For even a coarse grid, this approach is shown to have reliable estimation performance and leads to an IPP of good accuracy. Due to its parallelizable nature, the mixed search allows the two-stage strategy to be implementable in real time.
机译：研究了使用单个被动传感器（静态或匀速运动）的二维测量来估算在三维空间中用于冲击点预测（IPP）的推力/弹道内大气弹的状态的问题。弹丸的发射点（LP）的位置通常不可用，这可能会严重影响估算和IPP的性能。但是，如果已知LP的海拔高度，则可以从与地形图相交的第一条视线测量中以很小的误差获得发射位置。根据目标参数矢量的Fisher信息矩阵（FIM）分析可估计性，该参数确定其轨迹：初始发射角度（方位角和仰角），阻力系数和推力。缺乏关于LP高度的知识使问题变得更加困难，因为该高度是额外的未知目标参数，必须包含在需要可估计性分析的目标参数向量中。带有/不带有LP高度的FIM的最高等级可确保其具有可估计的目标参数。相应的Cramerer Rao下限量化了估计器的估计性能，该估计器在统计上是有效的，并且可以用于IPP精度评估。考虑到问题的固有非线性，可以通过使用适当的数值方法来找到目标参数向量的最大似然估计。提出了一种具有两个阶段的搜索策略，即混合（部分基于网格）搜索，然后是连续搜索。即使对于粗糙的网格，这种方法也显示出可靠的估计性能，并导致IPP具有良好的准确性。由于其可并行性，混合搜索使两阶段策略可以实时实施。

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