In this paper we present a new approach to solve the fuel-efficient powered descent guidance problem on large planetary bodies with no atmosphere (e.g., Moon or Mars) using the recently developed Theory of Functional Connections. The problem is formulated using the indirect method which casts the optimal guidance problem as a system of nonlinear two-point boundary value problems. Using the Theory of Functional Connections, the problem’s linear constraints are analytically embedded into a functional, which maintains a free-function that is expanded using orthogonal polynomials with unknown coefficients. The constraints are always analytically satisfied regardless of the values of the unknown coefficients (e.g., the coefficients of the free-function) which converts the two-point boundary value problem into an unconstrained optimization problem. This process reduces the whole solution space into the admissible solution subspace satisfying the constraints and, therefore, simpler, more accurate, and faster numerical techniques can be used to solve it. In this paper a nonlinear least-squares method is used. In addition to the derivation of this technique, the method is validated in two scenarios and the results are compared to those obtained by the general purpose optimal control software, GPOPS-II. In general, the proposed technique produces solutions of \(\mathcal {O}(10^{-10})\) accuracy. Additionally, for the proposed test cases, it is reported that each individual TFC-based inner-loop iteration converges within 6 iterations, each iteration exhibiting a computational time between 72 and 81 milliseconds, with a total execution time of 2.1 to 2.6 seconds using MATLAB. Consequently, the proposed methodology is potentially suitable for real-time computation of optimal trajectories.

在本文中，我们提出了一种新方法，利用最近开发的功能连接理论来解决没有大气层的大型行星体（例如月球或火星）上的燃油效率动力下降引导问题。该问题采用间接方法来表述，该方法将最优制导问题视为非线性两点边值问题系统。使用函数连接理论，问题的线性约束被分析地嵌入到函数中，该函数保持使用具有未知系数的正交多项式展开的自由函数。无论未知系数（例如，自由函数的系数）的值如何，约束总是在分析上得到满足，这将两点边值问题转换为无约束优化问题。该过程将整个解空间缩减为满足约束的容许解子空间，因此可以使用更简单、更准确、更快速的数值技术来求解。本文采用非线性最小二乘法。除了该技术的推导之外，该方法还在两种情况下进行了验证，并将结果与​​通用最优控制软件 GPOPS-II 获得的结果进行了比较。一般来说，所提出的技术产生\(\mathcal {O}(10^{-10})\)精度的解决方案。此外，对于所提出的测试用例，据报道，每个基于 TFC 的内循环迭代在 6 次迭代内收敛，每次迭代的计算时间在 72 到 81 毫秒之间，使用 MATLAB 的总执行时间为 2.1 到 2.6 秒。 因此，所提出的方法可能适用于最佳轨迹的实时计算。