通过函数联系理论的大型行星体上的燃料有效的动力下降制导
Fuel-Efficient Powered Descent Guidance on Large Planetary Bodies via Theory of Functional Connections
摘要：
在本文中，我们提出了一种新的方法来解决燃料效率的动力下降制导问题的大型行星机构，没有大气层（如月球或火星）使用最近发展的功能连接理论。采用间接法将最优制导问题转化为非线性两点边值问题。使用功能连接的理论，问题的约束分析嵌入到一个"约束表达式"，它保持一个自由函数，使用正交多项式与未知系数扩展。无论未知系数的值如何，约束都得到满足，从而将两点边值问题转化为无约束的最优化问题。这个过程将解投射到问题的容许子空间中，因此可以使用简单的数值技术（即本文使用非线性最小二乘法）。除了这种技术的推导，该方法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所获得的结果进行了比较。通常，所提出的技术产生$\mathcal{O}(10^{-10})$的解。此外，对于提出的测试用例，据报告，每个基于TFC的内部循环迭代在6次迭代内收敛，每次迭代在MATLAB传统实现中的计算时间在72到81毫秒之间。因此，所提出的方法是潜在的适合于板上实时生成的最佳轨迹。
英文摘要：
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. the 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 constraints are analytically embedded into a "constrained expression," which maintains a free-function that is expanded using orthogonal polynomials with unknown coefficients. The constraints are satisfied regardless of the values of the unknown coefficients which convert the two-point boundary value problem into an unconstrained optimization problem. This process casts the solution into the admissible subspace of the problem and therefore simple numerical techniques can be used (i.e. 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 $\mathcal{O}(10^{-10})$. 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 within the MATLAB legacy implementation. Consequently, the proposed methodology is potentially suitable for on-board generation of optimal trajectories in real-time.
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