Capturability of 3D PPN against lower-speed maneuvring target for homing phase

The capturability of two-dimensional (2D) 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 three-dimensional (3D) relative kinematics, the NOR method has not been applied to the capturability analysis of 3D PPN, which leads to the capturability discrepancy of 2D PPN and its 3D extension. Thanks to the 3D relative kinematic equation between the missile and target established in the rotating line of sight (LOS) coordinate system, the capturability of 3D PPN against the lower-speed arbitrarily maneuvering target for the homing phase is restudied by extending the NOR method of 2D PPN to the 3D space. The necessary and sufficient condition for the missile guided by 3D PPN to intercept this type of target is obtained. It is proven that the capturability of 3D PPN is identical with that of 2D PPN.