The trajectory estimation problem of a thrusting/ballistic object in three-dimensional (3-D) space has been previously solved with 2-D measurements (azimuth and elevation angles from a fixed passive sensor, either starting from the launch time or with a delayed acquisition) under the assumption of constant mass. However, since the mass decreases as the fuel burns, this should be accounted for. This paper investigates several approaches with different parameter vectors to solve the trajectory estimation and impact point prediction (IPP) with measurements starting after the launch time, i.e., delayed acquisition for both constant mass motion model and mass ejection motion model. For the mass ejection motion model, the mass ejection rate is an extra component of the parameter vector to be estimated. The invertibility of the Fisher information matrix (FIM) of the parameter vectors is also used to confirm the observability (estimability) of the system. The Cramer–Rao lower bound (CRLB) is the inverse of the FIM if it is invertible. The CRLB of the IPP is also derived. We develop the maximum likelihood estimator of the considered motion parameter vectors. Performance comparison between the models considered is given and the statistical efficiency of the best model is confirmed via simulation results.

中文翻译：

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具有延迟采集的单个固定无源传感器质量抛射的推力/弹道物体的估计

三维 (3-D) 空间中推进/弹道物体的轨迹估计问题先前已通过 2-D 测量（来自固定无源传感器的方位角和仰角，从发射时间开始或延迟收购）在恒定质量的假设下。但是，由于质量随着燃料燃烧而减少，因此应该考虑到这一点。本文研究了几种具有不同参数向量的方法，以解决轨迹估计和撞击点预测 (IPP)，并在发射时间之后开始测量，即恒定质量运动模型和质量抛射运动模型的延迟采集。对于质量抛射运动模型，质量抛射率是要估计的参数向量的额外分量。参数向量的 Fisher 信息矩阵 (FIM) 的可逆性也用于确认系统的可观察性（可估计性）。Cramer-Rao 下界 (CRLB) 是 FIM 的倒数，如果它是可逆的。IPP 的 CRLB 也是导出的。我们开发了所考虑的运动参数向量的最大似然估计器。给出了所考虑模型之间的性能比较，并通过仿真结果确认了最佳模型的统计效率。