Dynamical structures in the circular restricted three-body problem (CR3BP) are fundamental for designing low-energy transfers, as they aid in analyzing phase space transport and designing desirable trajectories. This study focuses on lobe dynamics to exploit local chaotic transport around celestial bodies, and proposes a new method for systematically designing low-energy transfers by combining multiple lobe dynamics. A graph-based framework is constructed to explore possible transfer paths between departure and arrival orbits, reducing the complexity of the combinatorial optimization problem for designing fuel-efficient transfers. Based on this graph, low-energy transfer trajectories are constructed by connecting chaotic orbits within lobes. The resulting optimal trajectory in the Earth--Moon CR3BP is then converted into an optimal transfer in the bicircular restricted four-body problem using multiple shooting. The obtained transfer is compared with existing optimal solutions to demonstrate the effectiveness of the proposed method.
Space exploration in cislunar space has attracted increasing interest due to the Artemis program [1] and the growing number of small satellite missions [2,3]. Cislunar missions have different requirements for fuel consumption and time of flight, which directly affect trajectory design. Recent missions such as Chang'e-3 [4,5], Artemis 1 [6,7], and Chandrayaan-3 [8] used fast transfer trajectories that require more fuel. In contrast, missions like Hiten [9,10], SMART-1 [11,12], GRAIL [13], CAPSTONE [14], and SLIM [15][16][17] adopted fuel-efficient transfers with longer time of flight. A key challenge in trajectory design is to find a variety of transfer options that meet specific mission criteria [18]. This is difficult because multi-body environments like cislunar space are highly sensitive to initial conditions, with spacecraft influenced by multiple celestial bodies simultaneously. Trajectory design in such environments must harness the chaotic dynamics [19,20].
Trajectory design methods in literature fall into two main categories: numerical and geometrical. Numerical approaches (e.g., Refs. [21][22][23][24]) identify transfer trajectories by solving optimization problems. By evaluating a large set of initial conditions, these methods can find a wide range of solutions for fuel consumption and time of flight, often yielding Pareto-optimal sets. Their drawbacks include the need for accurate initial guesses and high computational cost. Geometrical or dynamical systems approaches (e.g., Refs. [25][26][27][28][29][30][31]) use the natural dynamics described by dynamical systems theory to generate low-energy transfers. These approaches are limited to planar problems and often result in longer transfer times. The Genesis mission [32,33] was the first to demonstrate the practical value of this approach by designing its nominal trajectory using modern dynamical systems theory. Both approaches are valuable and complementary, but further development of the geometrical approach is crucial to address the key challenge in trajectory design. The geometrical approach identifies desirable trajectories by analyzing phase space transport, while the numerical approach focuses on local dynamics near given points. Although the numerical approach may imply underlying dynamical structures, the geometrical approach is generally more effective at providing a variety of trajectories with specific properties.
The geometrical approach produces low-energy transfers based on chaotic transport in natural flows, as described by the theories of lobe dynamics and tube dynamics (e.g., Refs. [34][35][36]). These theories utilize invariant manifolds associated with periodic orbits in the planar circular restricted three-body problem (CR3BP) [37]. The four-dimensional phase space in the planar CR3BP is often analyzed using a two-dimensional Poincaré map. If periodic orbits appear as points on a Poincaré map, their manifolds may reveal lobe dynamics; otherwise, the manifolds may describe tube dynamics. Thus, it is important to analyze the manifolds of periodic orbits with an appropriate Poincaré map.
Tube dynamics [25,38] describes global transport between two celestial bodies based on the stable and unstable manifolds of libration point orbits [34], especially around L 1 or L 2 in the planar CR3BP. These manifolds are called Conley-McGehee tubes or simply tubes [38]. The tubes around L 1 and L 2 provide a framework for understanding the dynamics in the planar CR3BP, as shown in Ref. [26]. For example, the natural orbit transitions of Jupiter comets are described using tube dynamics in the Sun-Jupiter system [39]. Tube dynamics also explains the mechanism of ballistic lunar transfers through Earth-Moon L 2 , enabling systematic design of such transfers [40]. One limitation of tube dynamics is that tubes associated with L 1 Lyapunov orbits do not help depart from the primary, as they move away from it in the CR3BP. Another limitation is the difficulty of applying this method to interplanetary transfers in the Solar System [36]. For example, tubes in the Sun-Earth and Sun-Mars planar CR3BP do not naturally intersect [36,41].
Lobe dynamics describes finer transport between regions around a single celestial body [34]. This theory was first developed to analyze fluid mixing in two-dimensional inviscid incompressible flows [42] and then extended to two-dimensional, area-and orientation-preserving maps [43]. In such maps, segments of the stable and unstable manifolds associated with periodic points may form small enclosed regions called lobes, which are key to transport by lobe dynamics. Within the theory of transport in dynamical systems [44,45], this idea has been used to analyze the chaotic transport of phase space volume between regions in various fields, such as fluid dynamics [46][47][48][49][50], geophysical flows [51][52][53], celestial mechanics [36,54], and chemical reaction dynamics [55,56]. Computational tools for lobe area and transp
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