Multi-stage air defense missile trajectory optimization using Gauss Pseudospectral Method

Gauss Pseudospectral Method(GPM) is employed to solve a fast trajectory optimization problem for a multi-stage air defense missile. The optimal solution maximizes the final velocity at the time when the missile locates at the predicted intercept point. The GPM is a collocation method where the states and controls are discretized through Lagrange interpolation, and the orthogonal collocation of the dynamics is performed at the Legendre-Gauss points. The original continuous-time two-point boundary value problem thus is discretized to a nonlinear programming. The state equations are not the same in different stages. The trajectory of the air defense missile is thus divided into different phases. The GPM is directly employed in each phase, and adjacent phases are linked together by imposing equality constrains on their boundary nodes to make sure that the position and velocity are continuous. The equality constrains are imposed on the terminal node to make the missile achieve the predicted intercept point. The path constrains are imposed to ensure the missile's vertical launch. The results presented in this paper show that the GPM is a viable approach The equality constrains and path constrains can be satisfied. The trajectory optimization not only completes fast, but also enjoys its high accuracy, which promotes the potential usage of GPM.