Particle Swarm Optimization-Based Costate Initialization for Minimum Time Transfers Between Cislunar Periodic Orbits

Low-thrust spacecraft, such as those using solar electric propulsion (SEP), are more efficient than traditional chemical propulsion systems due to their high specific impulse. This efficiency allows for larger payloads and less fuel carried onboard, which can reduce overall mission cost. This capability has generated more interest in low-thrust systems; however, the transfers for such systems are generally more complex and difficult to compute. The difficulty in computing optimal low-thrust transfers is primarily due to the associated limited thrust magnitude, which provides low control authority, requires long continuous applications of thrust, and often the traversal of many revolutions about one or more primary bodies. The methods employed to compute such transfers typically fall into the categories of direct and indirect methods, both of which generally require a quality initial guess for the unknown variables to successfully converge to a solution in this context. Direct methods transcribe the low-thrust optimal control problem into a large nonlinear programming (NLP) problem with a majority of the unknown (i.e., design) variables corresponding to a temporal discretization of the full state and control trajectories. As such, intuition about a given low-thrust transfer problem can often be employed to construct a guess for such variables, but the automation of this process can be difficult. Alternatively, indirect methods involve the formulation of a boundary value problem (BVP) through the application of variational calculus and Pontryagin’s maximum principle (PMP), relating the optimal controls to the state and costate variables. Therefore, the unknown variables of an indirect method primarily consist of state or costate variables at the temporal boundaries of the optimal control problem. Due to the abstract nature of the costate variables, intuition alone is generally insufficient to directly construct a quality guess for an indirect method. However, the small number of unknown variables associated with an indirect method lends well to the automated construction of an initial guess. Often, some form of stochastic initialization is used to construct an initial guess for an indirect method. A simple approach involves the realization of a guess from a uniform distribution. However, this method generally produces low-quality guesses of the unknowns and must typically be applied many times to find a guess that successfully converges to a solution of the BVP, a process that can be inefficient. As such, more sophisticated methods have been proposed in the literature that can stochastically generate a higher-quality guess such that only a few (if not just one) need to be generated to find a solution to the desired problem. One such approach of particular interest is referred to as particle swarm optimization (PSO)-based costate initialization, first proposed in [1,2] and later refined in [3]. In [3], PSO is employed to approximately solve an auxiliary problem to generate a quality guess for solving an optimal control problem with the indirect method​. The PSO algorithm operates within a constrained search space by considering transfer minimum radii constraints about primary bodies and minimum and maximum transfer time of flights, in addition to bounds on the design variables of the auxiliary problem​. The Jacobi integral of the final state of a candidate transfer is also considered in the auxiliary problem, which was found to improve the quality of the generated guesses [3]. However, the approach proposed in [3] was specifically developed for and only applied to low-thrust spiral problems (i.e., low-thrust transfers from Earth orbit to the cislunar regime). PSO-based costate initialization methods, in general, have also only been applied to a small set of problems in the literature [1,2]. In this work, the PSO-based costate initialization method proposed in [3] will be applied to compute transfers between periodic orbits of the cislunar regime in multiple unique transfer scenarios that have not been investigated previously. The method will be applied to such scenarios to investigate the applicability of the method to a different class of low-thrust transfer scenarios, i.e., to determine where it works well and where it falls short. In analyzing the results of these transfers, this work also aims to propose advancements to the PSO-based initialization strategy to further improve its applicability to the class of problems investigated (i.e., transfers between cislunar periodic orbits). The proposed advancements will be applied to the same transfer scenarios to facilitate the quantification of its improvement over existing methods in the literature. The PSO-based costate initialization methods investigated in this work will be applied to compute minimum time low-trust transfers between several unique L1 and L2 halo orbits and distant retrograde orbits (DROs). This will be done while considering a range of thrust magnitudes decreasing from a maximum thrust magnitude (1 N in most cases) across many trials in a Monte Carlo-like analysis to determine for which low-thrust magnitude a given costate initialization method begins to struggle in generating a guess that will successfully converge to a solution. It is expected that all methods investigated will produce guesses that converge to solutions with a high probability for the largest thrust magnitude considered over all scenarios investigated. However, each method investigated is expected to struggle at generating a guess that will converge to a solution for different thrust magnitudes across the different scenarios considers which will shed light on the applicability of each to computing solutions for this class of low-thrust transfer problems. [1] Jiang, F., Baoyin, H., and Li, J., “Practical Techniques for Low-Thrust Trajectory Optimization with Homotopic Approach,” Journal of Guidance, Control, and Dynamics, Vol. 35, 2012, pp. 1671-1676. [2] Hecht, G. R. and Botta, E. M., “Particle Swarm Optimization-Based Co-State Initialization for Low-Thrust Minimum-Fuel Trajectory Optimization,” Acta Astronautica, Vol. 211, 2023, pp. 416–430 [3] Hecht, G. R. and Botta, E. M., “Advancements in Particle Swarm Optimization-Based Costate Initialization for Spiral Trajectory Optimization in Cislunar Mission Design,” 2024 AIAA/AAS Astrodynamics Specialist Conference, Broomfield, CO., 2024