The optimal guidance method for fixed-time and fixed-point orbit injecting

This paper proposes an optimal guidance method which can give a certain orbit injection time, and satisfy the six terminal constraints of velocity and position at the same time. An optimal control problem of fixed-time and fixed-point orbit injection is established for the launch vehicle whose size of engine thrust and direction are controllable. The problem is proved to be solvable by introducing a new control variable, transforming the problem with equality constraints into control variable unconstrained problem, and deriving formulas. Numerical algorithm can be used to calculate seven co-state variables, then the size and direction of thrust is obtained. The simulations reveal that the guidance method this paper presented can put the rocket into the target orbit from the specified orbit injection point at the specified time.