Analytic study on extended trajectory shaping guidance system with first-order autopilot dynamics

Based on the time-to-go function with-nth power, the extended objective function is presented and the extended trajectory shaping guidance law is proposed for the lag-free guidance system. A third-order linear time-varying homogeneous differential equation is obtained when a first-order autopilot is introduced into the extended trajectory shaping guidance system and using the power series solution method, and the analytic solutions of position, velocity and acceleration are derived. According to the analytic results of position and velocity at final time, the analytic expressions of the miss distance and terminal impact angle error due to first-order autopilot dynamics are obtained. The analytic results of miss distance and terminal impact angle error are validated by the method of adjoint. The simulation and analytic results show that the miss distance and impact angle error induced by heading error and terminal impact angle approaches to zero when the missile guidance time is about 15 times of the autopilot lag time.