Analytical solution of generalized three-dimensional proportional navigation

Three second-order nonlinear differential equations that describe the motion of missiles guided by generalized three-dimensional proportional navigation are solved analytically without any linearization. Based on the introduction of unit angular momentum for the relative motion, a new moving coordinate system is proposed, instead of the conventional moving spherical coordinate system, to decouple the equations of motion so that relative distance and azimuths can be solved independently. It is hoped that this approach provide a systematic framework for analyzing three-dimensional guidance laws in an analytical way and bridges the mathematical gap between two-dimensional guidance laws and three-dimensional guidance laws in the nonlinear setup.