Analytical solution of three-dimensional realistic true proportional navigation

Abstract

True proportional navigation with varying closing speed is called realistic true proportional navigation, which is implemented in practice. Our main goal is to derive the complete solutions of three-dimensional realistic true proportional navigation for nonmaneuvering and maneuvering targets. Three coupled nonlinear second-order state equations describing the relative motion are solved analytically without any linearization for performance and trajectory analysis. Properties of three-dimensional realistic true proportional navigation such as capture region, range-to-go, time-to-go, and two aspect angles within spherical coordinates are all obtained in closed form. Furthermore, the two-player game between three-dimensional realistic true proportional navigation and three-dimensional ideal proportional navigation is investigated analytically in the pursuit-evasion scenario, where a realistic true proportional navigation guided missile is designed to pursue an ideal proportional navigation guided target. It is found that an ideal proportional navigation guided target is much harder to intercept than a realistic true proportional navigation guided target.