Analytical solution of three-dimensional realistic true proportional navigation

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 threedimensional 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. I. Introduction ROPORTIONAL navigation (PN) for short-range tactical misP siles is the optimal interceptive law in the sense of producing zero miss distance with the least integral square control effort. PN has been studied since the 1940s. During the four decades that followed, proportional navigation has been explored in many different ways, such as true proportional navigation, pure proportional navigation (PPN), generalized proportional navigation, realistic true proportional navigation (RTPN), and ideal proportional navigation It has long been recognized that there exists a significant difference in the way in which PN guidance law is analyzed and in the way in which it is implemented. Most analytical studies of PN assume that the closing velocity in the PN guidance law is constant, whereas in realistic situations, the instantaneous closing speed may be continuously estimated or measured using devices such as homing seekers with Doppler radar. To remove this difference, RTPN, which adapts to varying closing speed, has recently been proposed. Performance and trajectory analysis of two-dimensional RTPN was studied by