Three-dimensional air-to-air missile trajectory shaping

Abstract

Open-loop range/energy/time optimal trajectories are synthesized for a three-dimensional point-mass model of a boost-sustain-coast air-to-air missile (AAM) in atmospheric flight. Boundaries of attainable sets for various fixed flight times are obtained by numerically solving the associated two-point boundary value problem (TPBVP) implied by the first-order necessary conditions of optimal control. The attainable set provides insight into the range capabilities of the missile in all directions in the down range-cross range plane for prescribed end-game energy requirements. Necessary and sufficient conditions for optimality are checked for candidate extremals. A new governing matrix differential equation and suitable boundary conditions are derived to check for conjugate points along regular or regularized extremals. This new matrix differential equation overcomes the difficulty of standard matrix Riccati equations for conjugate point testing, which have components whose values go to infinity even in the absence of conjugate points. The open-loop solutions presented in this study can be utilized to generate a neighboring near-optimal guidance scheme.