Cislunar Linear Transfer: CR3BP Linearization

• Cislunar Linear Transfer is a method for designing optimized low-energy trajectories in the Earth–Moon system via linearized CR3BP dynamics and invariant manifold structures.
• It employs techniques such as higher-order Taylor expansions, lobe dynamics, and graph-based search to efficiently explore and optimize phase-space transport.
• Incorporating four-body refinement ensures the robustness of transfers by accounting for solar perturbations, leading to Pareto-optimal performance in Δv and time-of-flight.

A cislunar linear transfer refers to the design and optimization of low-energy spacecraft trajectories in the Earth–Moon system, leveraging the underlying dynamical structures of the Circular Restricted Three-Body Problem (CR3BP) and related models. Linearization at cislunar libration points (L₁, L₂) enables analytic and computational methods for transfer design, while tools from lobe dynamics, graph-based search, and higher-order Taylor expansions support efficient exploration and optimization of phase-space transport. These advances underpin modern strategies for mission planning, including ballistic capture, periodic orbit insertion, and robust refinement under four-body perturbations.

1. Linearization of the CR3BP at Cislunar Libration Points

The CR3BP describes the motion of a massless particle under the gravitational influence of two primaries (e.g., Earth and Moon) in a co-rotating reference frame. The equations of motion are given by

$$\ddot x - 2\dot y = \frac{\partial U}{\partial x}, \quad \ddot y + 2\dot x = \frac{\partial U}{\partial y},$$

where $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$, $r_1 = \sqrt{(x + \mu)^2 + y^2}$, $r_2 = \sqrt{(x - 1 + \mu)^2 + y^2}$, and $\mu \approx 0.01215$ for the Earth–Moon system.

Linearization about an equilibrium point $(x_0, y_0)$ (L₁ or L₂) is performed by expanding the vector field as $\mathbf{x} = \mathbf{x}_L + \Delta \mathbf{x}$, leading to the variational system

$\dot{\delta X} = A\,\delta X,$

where $A$ is the Jacobian matrix evaluated at $\mathbf{x}_L$, incorporating second derivatives of $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$0. The characteristic equation yields eigenvalues that indicate local stable, unstable, and oscillatory directions, directly tied to the phase-space structure supporting transfer design via invariant manifolds (Hiraiwa et al., 19 Feb 2026, Anoè et al., 7 Jul 2025).

Higher-order Taylor expansions (up to fourth order or higher) about $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$1 or $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$2 further enhance the local fidelity of the dynamics: $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$3 Automatic differentiation (AD/DA) tools can systematically construct the coefficient arrays for such high-order expansions (Anoè et al., 7 Jul 2025).

2. Periodic Orbits, Manifolds, and Lobe Dynamics

Families of periodic orbits (Planar Lyapunov, halo, distant retrograde) in the CR3BP provide the backbone for low-energy transfer strategies. Newton-Raphson differential correction, leveraging symmetry constraints and Jacobi integral conservation, efficiently computes periodic orbits at fixed energy (Hiraiwa et al., 19 Feb 2026, Anoè et al., 7 Jul 2025).

The monodromy matrix (state transition matrix over one period) delivers eigenstructure for parameterizing local stable/unstable manifolds. Orbits initialized along these directions (via $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$4) are integrated to trace tubular phase-space structures, which guide long-term transport in the cislunar phase space.

Lobe dynamics provides a geometric framework for analyzing chaotic transport. On a chosen Poincaré section, intersections of stable/unstable manifolds define primary intersection points (pips), which in turn bound lobes (regions of phase space facilitating transport). The turnstile mechanism describes how these lobes mediate transitions between resonance zones under the Poincaré map, with pip detection and lobe quantification conducted via spline parameterizations and robust centroid/radius extraction (Hiraiwa et al., 19 Feb 2026).

3. Transfer Trajectory Construction: Graph-Based and Differential Algebraic Methods

A major advance in transfer design is the formalization of possible transport paths as weighted graphs, whose nodes comprise periapsis points, lobe centroids, and manifold intersections. Edges correspond to either "natural arcs" (zero-Δv, purely dynamical evolution through lobes) or "controlled arcs" (impulsive maneuvers). Weights encode cost metrics such as total Δv and time-of-flight.

Graph search algorithms—Dijkstra, A*, or exhaustive pruning—identify optimal or near-optimal sequences by minimizing the sum of arc weights, often enforcing constraints to maintain contiguous lobe‐sequence traversal and thus guarantee the exploitation of chaotic tubes. This combinatorial framework dramatically reduces the search complexity of multi-impulse trajectory optimization (Hiraiwa et al., 19 Feb 2026).

For optimization of transfers linking ballistic captures to periodic orbit families, a high-order Taylor expansion (built on DA techniques) supports the construction of polynomial parameterizations (abacus maps) of orbit families. These maps, together with sensitivity analyses, feed into a bi-impulsive shooting framework. The transfer is formulated as the minimization of

$$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$5

subject to boundary conditions matching the BC state and the desired periodic orbit via analytic maps and flow propagation over transfer time. The analytic Jacobian accelerates solver convergence, with DA sensitivities reducing computational cost by 50–80% relative to finite-difference approaches (Anoè et al., 7 Jul 2025).

4. Four-Body Refinement and Sensitivity to Perturbations

Because the real Earth–Moon–Sun system cannot be captured by the CR3BP alone, transfers of practical interest are refined in the bicircular restricted four-body problem (BCR4BP). This model augments the CR3BP with a periodically varying solar gravitational field, yielding non-autonomous equations: $$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$6 with

$$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$7

Multiple-shooting optimization decomposes the transfer into temporal segments, enforcing continuity and impulsive corrections at segment boundaries. The objective minimizes the sum of Δv for all corrections. This approach ensures robust transfer feasibility even under significant solar perturbation and allows the exploitation of resonance transport mechanisms identified in lobe dynamics (as evidenced by plateaus in semi-major axis evolution and the persistence of FTLE ridges) (Hiraiwa et al., 19 Feb 2026).

5. Performance Metrics and Benchmark Comparisons

Key applications demonstrate that lobe-dynamic and graph-based transfer design, followed by refinement in the BCR4BP, yields competitive or Pareto-optimal trajectories among known low-energy interior cislunar transfers. The method shows the following benchmark results (Hiraiwa et al., 19 Feb 2026):

Transfer Scenario	ΔV [m/s]	TOF [days]	Reference
LEO→LLO (CR3BP)	4274.67	191.9	(Hiraiwa et al., 19 Feb 2026)
LEO→LLO (BCR4BP, θ_s=0)	3832.61	193.3	(Hiraiwa et al., 19 Feb 2026)
Sweetser (1991)	3726	–	Literature
Topputo (2013) Hohmann	3954	5	Literature
Yagasaki (2004)	3925	31	Literature

The low-energy transfer designed via lobe dynamics, graph search, and BCR4BP refinement sits on the Pareto front with respect to Δv and time-of-flight. Notably, transfers can utilize a sequence of small impulsive corrections (<9 m/s) in combination with major injections, further highlighting the efficacy of leveraging phase-space transport structures. Ballistic capture to planar or spatial periodic orbit families via bi-impulsive optimization achieves satisfying cost and rapid algorithmic convergence due to high-order analytic sensitivities (Anoè et al., 7 Jul 2025).

6. Summary and Impact on Cislunar Mission Design

The modular sequence—linearized local analysis at libration points, periodic orbit and manifold computation, lobe and turnstile extraction on Poincaré maps, graph-based pathfinding, BCR4BP refinement, and phase-space diagnostic validation—establishes a comprehensive toolkit for designing robust, low-energy cislunar transfers. These methods support current and future lunar missions by enabling both ballistic capture and periodic orbit insertion, delivering reliable initial guesses and detailed cost profiles suitable for rapid adaptation to mission needs. The analytic, high-order approaches outlined in contemporary work provide both conceptual clarity for constructing transfer strategies and algorithmic efficiency for operational implementation (Hiraiwa et al., 19 Feb 2026, Anoè et al., 7 Jul 2025).