Analytical solutions for longitudinal-plane motion of hypersonic skip-glide trajectory

The analytical solutions for longitudinal-plane motion of hypersonic skip-glide trajectory are proposed by perturbation theory. First, the analytical solution of downrange distance is approximated by the outer solutions of a two-timescale singular perturbation system where the solution error is less than 2%. Then, the zeroth-order solutions and first-order corrections of altitude and flight-path angle are developed using regular perturbation techniques which could decompose the coupled and nonlinear system into a sequence of linear problems to solve in order. Because the first-order system for altitude and flight-path angle is a linear time-varying system that cannot be solved by traditional method such as Laplace transform, two methods are put forward to simplify the system: spectral decomposition method (SDM) and linear transformation method (LTM). SDM assumes the system to be time invariant and then uses spectral decomposition to solve it. LTM introduces a small compensation into the system such that the system can be decoupled into two independent first-order differential equations by linear transformation. Then, by combining the advantages of the two kinds of solutions, more accurate integrated analytical solutions are proposed. The simulation results show that the analytical solutions are in good agreement with the numerical solutions. In addition, the computation time of the analytical solutions is about two orders of magnitude less than that of the numerical solutions.