The capturability of the two-dimensional (2-D) pure proportional navigation (PPN) guidance law against lower speed arbitrarily maneuvering target for homing phase had been thoroughly analyzed by using the nonlinear output regulation (NOR) method before. However, due to the complexity of the 3-D relative kinematics, the NOR method has not been applied to the capturability analysis of the 3-D PPN, which leads to the capturability discrepancy of the 2-D PPN and its 3-D extension. Thanks to the 3-D relative kinematic equation between the missile and target established in the rotating line-of-sight coordinate system, the capturability of the 3-D PPN against the lower speed arbitrarily maneuvering target for the homing phase is restudied by extending the NOR method of the 2-D PPN to the 3-D space. The necessary and sufficient condition for the missile guided by the 3-D PPN to intercept this type of target is obtained. It is proven that the capturability of the 3-D PPN is identical with that of the 2-D PPN.

中文翻译：

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3D PPN 对低速机动目标的捕获能力，用于归航阶段

之前已经使用非线性输出调节（NOR）方法彻底分析了二维（2-D）纯比例导航（PPN）制导律对寻航阶段低速任意机动目标的可捕获性。然而，由于 3-D 相对运动学的复杂性，NOR 方法尚未应用于 3-D PPN 的可捕获性分析，这导致了 2-D PPN 及其 3-D 扩展的可捕获性差异. 由于在旋转视距坐标系中建立了导弹与目标之间的 3-D 相对运动学方程，通过扩展导引阶段的 3-D PPN 对低速任意机动目标的捕获能力进行了重新研究。 2-D PPN 到 3-D 空间的 NOR 方法。得到了3-D PPN制导导弹拦截此类目标的充要条件。已证明 3-D PPN 的可捕获性与 2-D PPN 的捕获能力相同。