Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the double-homotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to a halo orbit is presented to substantiate the effectiveness of the technique.

Convex optimization is of great interest as an efficient and reliable solver in the field of aerospace engineering. Existing research often focuses on developing proper convexification techniques to handle aerospace problems within the convex optimization framework, with limited attention to further efficiency enhancements in convex optimization solving, such as by reducing the optimization problem size. Motivated by the real-time requirements of onboard aerospace applications, this paper presents a unified framework for reduced space sequential convex programming formulations, emphasizing a significant reduction of optimization variables and constraints. The primary idea is to employ an iterative scheme to explicitly approximate the dynamic implicit function and subsequently eliminate the explicitly defined state variables and dynamic equality constraints, thereby constructing a sequence of reduced space optimization subproblems to be solved iteratively. Within the proposed framework, a family of reduced space sequential convex programming methods with different performance-complexity tradeoffs are developed by virtue of the fixed-point iteration, Newton iteration, damped Newton iteration, and simplified Newton iteration. Numerical simulations for a minimum-fuel rocket landing problem inside atmosphere are conducted to demonstrate the performance of the developed methods.

This paper proposes an enhanced approach to sequential convex programming for optimizing onboard trajectories by incorporating non-uniform rational basis splines (NURBS) curves. The contributions of the proposed method are threefold. Firstly, it effectively reduces the number of optimization variables by utilizing NURBS curves as representations for control variables. Secondly, the convex-hull property of NURBS curves enables the replacement of the control inequality constraints with those defined by NURBS control points, leading to a significant reduction in the overall number of constraints. Lastly, the integration of the homotopy approach with NURBS curves offers increased flexibility in shaping the curves and facilitates the solution of optimization problems involving bang-bang optimal control. Numerical demonstrations are provided to illustrate the effectiveness and efficiency of the proposed approach for onboard applications.