In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

This manuscript investigates the use of a reinforcement learning method for the guidance of launch vehicles and a computational guidance algorithm based on a deep neural network (DNN). Computational guidance algorithms can deal with emergencies during flight and improve the success rate of missions, and most of the current computational guidance algorithms are based on optimal control, whose calculation efficiency cannot be guaranteed. However, guidance-based DNN has high computational efficiency. A reward function that satisfies the flight process and terminal constraints is designed, then the mapping from state to control is trained by the state-of-the-art proximal policy optimization algorithm. The results of the proposed algorithm are compared with results obtained by the guidance-based optimal control, showing the effectiveness of the proposed algorithm. In addition, an engine failure numerical experiment is designed in this manuscript, demonstrating that the proposed algorithm can guide the launch vehicle to a feasible rescue orbit.

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