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.

中文翻译：

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基于功能连接理论的运载火箭燃料优化上升轨迹问题

在这项研究中，我们开发了一种基于功能连接理论 (TFC) 的方法来解决运载火箭上升阶段的燃料优化问题。首先使用间接方法将问题转化为非线性两点边值问题（TPBVP）。然后，使用称为 TFC 的函数插值技术，将问题的约束解析嵌入到函数中，并将 TPBVP 转换为包含未知系数正交多项式的无约束优化问题。这个过程有效地减少了解的搜索空间，因为原来的有约束问题转化为无约束问题，因此无约束表达式的未知系数可以用简单的数值方法求解。最后，通过与通用非线性最优控制软件GPOPS-II和传统的间接数值方法进行对比，验证了所提出的算法。结果表明，所提出的算法对较差的初始值具有鲁棒性，并且在 MATLAB 实现中可以在不到 300 ms 的时间内求解。因此，所提出的方法有可能实时生成最佳的机载轨迹。