Numerical solution of optimal trajectory for hypersonic vehicle with constraints

In order to obtain the precise optimal reentry cruise trajectory for a hypersonic vehicle with constraints, a co-state curve estimation method is developed based on the suboptimal results of the optimal control problem computed by the direct method. Using this technique, not only the initial values but also the time history of the co-states can be precisely estimated. Furthermore, when path constraints existed in the system and the co-state curves are not continuous, the jumps of the co-states can be worked out. According to the characteristics of the hypersonic vehicle minimum fuel trajectory, the suboptimal results are calculated by sequential quadratic programming(SQP), and then the co-state initial values and jumps are guessed. After the two point boundary value problem(TPBVP) of the system is established based on the maximum principle, the optimal results are obtained. Simulation results indicate that under the constraint of a heat current of 300 W/cm² the optimal results save about 2.4% fuel over the steady solutions. Meanwhile, the difficulties of the TPVBP are reduced with the accurate co-state initial values and jumps provided by this method, so that the optimal results can be gained rapidly.