Capturability of 3D PPN Against Lower-Speed Maneuvering Target for Homing Phase

Abstract

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. Index Terms-3D PPN, capturability, nonlinear output regulation method, lower-speed maneuvering target I. INTRODUCTION OR airborne missiles, pure proportional navigation (PPN) guidance law is broadly recognized as one of the most robust and powerful guidance laws [1], and its capturability has been a major research issue for recent decades [2]~[10]. Endoatmospheric targets are commonly categorized into lower-speed and higher-speed targets 1 . For the class of lower-speed targets, Guelman [2]~[4] analyzed the capturability of PPN by using a qualitative method; Becker [5] obtained the closed-form solution of PPN based on the expansion theory of meromorphic function; Ghawghawe and Ghose [6] extended Guelman's method to the capturability analysis of PPN against the maneuvering target with a time-varying normal acceleration; Ha et al. [7] proposeda Lyapunov-like approach to analyze the performance of PPN against a randomly maneuvering target for the homing phase where the missile is initially flying toward the target.The above research is confined to two-dimensional (2D) space. Considering the cross coupling effect of pitch and yaw planes of missile, Song and Ha [8] extended the Lyapunov-like approach to the three-dimensional (3D) space, and analyzed the capturability, intercept time, line of sight rate convergence, and maximum commanded acceleration requirement of 3D PPN against lower-speed randomly maneuvering targets under the condition that the missile was initially flying toward the target. Based on the Lyapunov-like approach, Oh and Ha [9] proved that the missile guided by 3D PPN can always intercept a lower-speed arbitrarily maneuvering target with limited normal acceleration, when the navigation gain is large enough and the initial relative range is smaller than a certain value, no matter the missile is flying toward or away from the target initially.The prior research only focuses on the case that the missile-to-target range is doomed to be strictly decreasing after a finite time and excludes the case when the relative range has a f...

Performance of 3-D PPN Against Arbitrarily Maneuvering Target for Homing Phase

“…Later, Oh [25] proved that the capture condition of 2D PPN against arbitrarily maneuvering target is a large-than-one navigation gain, when the missile is with speed advantage. Recently, K. B. Li et al [26] extended Oh's result to the case of 3D PPN. According to our knowledge, this is the most general result about PPN's capturability thus far.…”

“…It is further assumed that the earth is nonrotating. Note that, these assumptions are generally accepted in the performance analysis of PPN guidance [4]~ [7], [21]~ [26].…”

“…According to our knowledge, this is the most general result about PPN's capturability thus far. However, in [24]~ [26], it is not strictly required that the closing speed between the missile and target is always negative. Hence, the required commanded acceleration of PPN to guarantee the capture of target might become extremely large.…”

An Improvement in Three-Dimensional Pure Proportional Navigation Guidance

“…The capturability condition of A10) is conservative: the condition implies that the missile flies towards to the target at the beginning of homing. Note that it is straightforward to find more general condition following [18] or [20]. However, since this is not the main scope of this study and derivation becomes lengthy, we limit our discussion on the capturability to Theorem 2.…”

“…Based on the Lyapunov-like approach [15]- [17], Oh and Ha [18], [19] presented the capture condition and the upper-bound of commanded acceleration of PPN against an arbitrarily manoeuvring target. Recently, K. B. Li et al [20] restudied the results of [17]- [19] and obtained more general conditions.…”