Traditionally, fuel-optimal problems for low-thrust spacecraft transfers are connected with energy-optimal problems to increase the convergence possibility of indirect methods. However, for low-thrust many-revolution missions, it is still hard to resolve the energy-optimal problems in a fast and reliable manner due to the strong nonlinearity. To address this issue, a warm-start multihomotopic optimization approach is proposed in this article, wherein the Sundman transformation and multihomotopic techniques are used to connect the energy-optimal problems to linear ones with analytical solutions. This article focuses on the following three contributions. First, the energy-optimal control problems are transformed into true longitude-depended energy-optimal problems based on the homotopy to dynamical model and Sundman transformation, and the equivalence relations of the solutions before and after transformation are obtained analytically. Second, the problems are further connected with time-free energy-optimal problems based on a homotopy to the time constraint. Third, the time-free problems are linearized near a nominal trajectory, and the corresponding analytical solutions are obtained. Starting with the analytical solutions, the algorithm can gradually iterate back to the solutions of the original energy- and fuel-optimal problems. Since the whole transformation process is lossless (optimality is preserved under the transformation process), the developed warm-start multihomotopic algorithm enjoys the advantages on reliable convergence and high computational efficiency. Numerical simulations of Earth-orbit transfer missions from GTO to GEO are conducted by comparing with traditional methods, and the results are given to substantiate the effectiveness of the proposed warm-start multihomotopic method.

中文翻译：

---

低推力多转轨迹的热启动多同伦优化

传统上，低推力航天器转移的燃料优化问题与能量优化问题相联系，以增加间接方法的收敛可能性。然而，对于低推力多转任务，由于强非线性，仍然难以快速可靠地解决能量优化问题。为了解决这个问题，本文提出了一种热启动多同伦优化方法，其中使用 Sundman 变换和多同伦技术将能量优化问题与具有解析解的线性问题联系起来。本文重点介绍以下三个贡献。首先，基于动力学模型的同伦和Sundman变换，将能量最优控制问题转化为真正的经度相关能量最优问题，并解析得到变换前后解的等价关系。其次，这些问题进一步与基于时间约束同伦的无时间能量最优问题相关联。第三，将无时间问题在标称轨迹附近线性化，并获得相应的解析解。从解析解开始，算法可以逐渐迭代回原始能源和燃料优化问题的解。由于整个变换过程是无损的（在变换过程中保持最优性），所开发的热启动多同伦算法具有收敛可靠和计算效率高等优点。