Abstract This paper presents a convex optimization-based approach to efficiently obtain the numerical solution of the fuel-optimal powered descent problem with free final-time and path constraints. To avoid guessing the final-time, we propose to choose altitude, instead of time, as the independent variable in the system dynamics. This selection also brings great convenience in incorporating the glide-slope and thrust direction constraints in which the bounds can be altitude-dependent. Then, the formulated optimal control problem is converted into a convex problem via appropriate convexification techniques, such as the nonlinearity-kept & linearization approach and relaxation, etc. Relaxation is a critical technique, but analyzing its validity is generally very challenging, especially when path constraints are present. In this paper we can prove that the relaxation used is valid. Next, we discretize the convex problem and apply successive convex optimization to get the solution of the original problem. Nevertheless, in order to obtain the solution with high accuracy and low computational cost, we propose a new strategy of selecting nonuniform discretized points plus the Runge-Kutta 4th order or trapezoidal discretization method. Numerical results are provided to show the effectiveness and high efficiency of the proposed method in solving the powered descent problem and reveal an interesting structure of the optimal thrust magnitude profile when path constraints become active.

中文翻译：

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具有自由最终时间和路径约束的燃料优化动力下降引导

摘要 本文提出了一种基于凸优化的方法，以有效地获得具有自由最终时间和路径约束的燃料优化动力下降问题的数值解。为了避免猜测最终时间，我们建议选择高度而不是时间作为系统动力学中的自变量。这种选择也为结合下滑道和推力方向约束带来了极大的便利，其中边界可以依赖于高度。然后，通过适当的凸化技术，如非线性保持和线性化方法和松弛等，将公式化的最优控制问题转换为凸问题。松弛是一项关键技术，但分析其有效性通常非常具有挑战性，特别是当路径存在约束。在本文中，我们可以证明所使用的松弛是有效的。接下来，我们将凸问题离散化并应用连续凸优化来获得原始问题的解决方案。尽管如此，为了获得高精度和低计算成本的解决方案，我们提出了一种新的选择非均匀离散点的策略，加上 Runge-Kutta 四阶或梯形离散化方法。提供的数值结果表明了所提出的方法在解决动力下降问题方面的有效性和高效率，并揭示了当路径约束变得活跃时最佳推力大小分布的有趣结构。为了获得高精度和低计算成本的解决方案，我们提出了一种新的选择非均匀离散点的策略，加上 Runge-Kutta 四阶或梯形离散化方法。提供的数值结果表明了所提出的方法在解决动力下降问题方面的有效性和高效率，并揭示了当路径约束变得活跃时最佳推力大小分布的有趣结构。为了获得高精度和低计算成本的解决方案，我们提出了一种新的选择非均匀离散点的策略，加上 Runge-Kutta 四阶或梯形离散化方法。提供的数值结果表明了所提出的方法在解决动力下降问题方面的有效性和高效率，并揭示了当路径约束变得活跃时最佳推力大小分布的有趣结构。