Three-dimensional united biased proportional navigation law for interception of maneuvering targets with angular constraint

Studying a guidance law, which can intercepts not only high-speed targets, but also low-speed targets with impact angle constrains, is a challenging task. In this paper, a new guidance law called united biased proportional navigation (UBPN) is proposed. The guidance law uses time-varying bias and navigation ratio, and incorporates the advantages of head-on and head-pursuit engagement. UBPN with positive time-varying navigation ratio can intercept low-speed targets which is a head-on engagement, while with negative time-varying navigation ratio can intercept high-speed targets, which is a head-pursuit engagement. To begin with, planar UBPN guidance law is proposed and exact time-varying bias is derived. Moreover, the solution to the singularity in time-varying navigation ratio is given. Secondly, in order to realize the guidance law in three-dimensional space, we derive the relation between two-dimensional acceleration and three-dimensional acceleration, and also give the solution for calculating the impact angle from two-dimensional surface into three-dimensional space. Finally, the simulation results of UBPN compared with proportional navigation, retro-proportional navigation, and bias proportional navigation guidance law are shown. It is demonstrated that the miss distance and impact angle error can meet the guidance law requirements. Capture region of UBPN is given by simulation, which indicates that it is almost two times than that of the guidance law in the head-on engagement for interception of high-speed target.