Transformer-Informed Trajectory Optimization for Relative Motion in Cislunar Orbits

INTRODUCTION

• Cislunar RPOD poses non-convex OCPs with constraints such as keep-out zones and approach ellipsoids, while Earth–Moon dynamics are more nonlinear and chaotic.Passive-safety constraints matter because lunar-orbit debris may remain in orbit or strike the lunar surface.Source: INTRODUCTION · pp. 1–2
• ART had previously been used for spacecraft RPOD only with perturbed two-body problems and linear relative-motion models, leaving its ability to handle cislunar chaos unproven.ART generates near-optimal states and control inputs as warm-starts for sequential convex programming (SCP).Source: INTRODUCTION · p. 3
• ART-TWIN adds forward-backward dual-arc trajectory generation to ART to support reliable convergence in dynamically chaotic regimes.The approach draws on forward-backward shooting methods used to reduce sensitivity in interplanetary trajectory optimization.Source: INTRODUCTION · p. 3
• ART-TWIN extends ART to cislunar RPOD with CR3BP nonlinear relative dynamics and is demonstrated at perilune, where the dynamics are most challenging.The paper describes this as ART's first extension beyond Earth orbit and its first use with a nonlinear dynamics model for trajectory generation.Source: INTRODUCTION · p. 3
• ART-TWIN ran 1.59x faster in median runtime than a convex relaxation warm-start, while original ART failed to converge via SCP in over 60% of the same test instances.The paper reports comparable optimality performance for ART-TWIN and the convex relaxation warm-start.Source: INTRODUCTION · p. 3

Circular Restricted Three-Body Problem

• The CR3BP models two spacecraft in the Earth–Moon system under the gravity of two primaries, treated as point masses in circular orbits about their common center of mass.The analysis uses nondimensional units and a synodic rotating frame.Source: Circular Restricted Three-Body Problem · p. 4
• The study uses a Moon-centered synodic frame and a chief-centered LVLH frame; the chief is depicted on a Near-Rectilinear Halo Orbit.The LVLH frame is anchored to the Moon, and the deputy's relative position is expressed with respect to the chief.Source: Circular Restricted Three-Body Problem · p. 4, Figure 1 · p. 4

Nonlinear Cartesian Relative Dynamics

• The nonlinear Cartesian relative dynamics are derived in an LVLH frame anchored to the Moon.The passage identifies the Earth-relative-to-Moon and chief-relative-to-Moon position vectors and CR3BP mass parameter used in the equations.Source: Nonlinear Cartesian Relative Dynamics · p. 5

Linearized Cartesian Relative Dynamics

• Linearizing gravitational acceleration about the chief’s LVLH position yields an LTV state-space model, with x representing the deputy’s Cartesian position and velocity.The model is written as ẋ = A(t)x.Source: Linearized Cartesian Relative Dynamics · p. 5
• The linearized model is expected to perform acceptably at apolune, but not at perilune.Source: Linearized Cartesian Relative Dynamics · p. 5

Autonomous Rendezvous Transformer

• ART autoregressively generates trajectories by predicting each step, using tokenized trajectory information to model states, controls, and performance metrics.Source: Autonomous Rendezvous Transformer · p. 5
• A tokenized trajectory contains states, control actions, reward-to-go, and constraint-to-go at each discretized time step.Reward-to-go is the cost from the current epoch to the terminal epoch; constraint-to-go counts steps with violations over that interval.Source: Autonomous Rendezvous Transformer · p. 6
• ART is designed to assist fixed-time, non-convex OCPs with prescribed initial and terminal conditions and dynamics that may be linear or nonlinear.Source: Autonomous Rendezvous Transformer · p. 6

Dataset Generation

• The dataset contains N_d trajectories obtained by repeatedly solving diverse instances of the optimization problem via SCP.Source: Dataset Generation · p. 6
• Each solved trajectory sample contains state and action time series, and randomized boundary conditions provide diversity across samples.The OCP parameters can be set for a designed mission scenario.Source: Dataset Generation · p. 6
• Exposure to near-optimal trajectories with varied initial conditions and performance metrics enables inference at boundary conditions outside the training set.The passage states that performance is maintained robustly for those conditions.Source: Dataset Generation · p. 6

Model Training

• ART training uses teacher forcing, minimizing a loss over batches of M trajectories using predicted states and control actions.Source: Model Training · p. 7

Model Inference

• During inference, dynamics-informed ART predicts controls autoregressively, propagates states through the dynamics model, and feeds updated state and performance metrics back into the model.The resulting trajectory warm-starts SCP, which enforces feasibility and hard constraint satisfaction in its final solution.Source: Model Inference · p. 7

Forward-Backward Dual-Arc Generation

• ART-TWIN generates a forward arc from the initial state and a backward arc from the terminal state, propagating both to a common midpoint breakpoint.The two arcs are patched and supplied to SCP as an initial guess.Source: Forward-Backward Dual-Arc Generation · pp. 7–8
• The dual-arc scheme shortens each generated arc and fixes both boundary states, while leaving a defect between the arcs for SCP to resolve.The paper presents this defect as easier to patch than terminal-state error in forward-only generation.Source: Forward-Backward Dual-Arc Generation · pp. 8–9
• An added dt token provides propagation-time information, enabling one ART-TWIN inference model to generate trajectories in both time directions and support varying discretizations.Source: Forward-Backward Dual-Arc Generation · p. 9

Applying ART-TWIN to Cislunar Rendezvous

• ART-TWIN is tested against a convex rendezvous warm-start that omits the keep-out-zone constraint and is also compared with ART-0.The evaluation considers terminal-state accuracy, constraint violations, control cost, runtime, convergence, and efficiency.Source: Applying ART-TWIN to Cislunar Rendezvous · p. 9

Problem 1: Convex, Linear Rendezvous Problem

• The convex relaxation is a fuel-optimal rendezvous problem, and its solution supplies both the SCP-trajectory training dataset baseline and the comparison warm-start.Source: Problem 1: Convex, Linear Rendezvous… · pp. 9–10
• The convex problem’s discrete dynamics use a state transition matrix and control-input matrix to propagate the state between time steps.The transition matrix is computed by expanding the matrix exponential series to fifth order.Source: Problem 1: Convex, Linear Rendezvous… · pp. 9–10

• The nonlinear, nonconvex OCP provides the trajectories used to train ART-TWIN and is solved iteratively with an optimization protocol such as SCP.Source: Problem 2: Non-Convex, Nonlinear… · p. 10
• The discrete-time dynamics constrain each next state using the nonlinear cislunar relative-motion function and the control input.Source: Problem 2: Non-Convex, Nonlinear… · p. 10
• Passive safety uses approach and keep-out-zone ellipsoids, with the active constraint switching from the approach sphere to the keep-out zone at step ngate.The constraints enforce free-drift safety for nsafe steps within a predefined safety horizon.Source: Problem 2: Non-Convex, Nonlinear… · p. 10

Sequential Convex Programming

• SCvx* iteratively updates the trust region and penalty weights; non-convergence remains possible, and the initial guess can affect the local optimum reached.The solver penalizes violations of linearized nonconvex constraints and normalizes variables relative to the initial guess.Source: Sequential Convex Programming · p. 10
• Each convexified subproblem linearizes passive-safety constraints around the previous iteration’s deputy states while retaining nonlinear dynamics propagation.The resulting converged solution matches nonlinear ground-truth propagation within machine error tolerances.Source: Sequential Convex Programming · p. 11
• This work applies dual- or multi-arc generation to ART rather than to the SCP process, although the latter remains a possible avenue for investigation.Source: Sequential Convex Programming · p. 11

RESULTS AND ANALYSIS

• The experiments assess ART-TWIN for cislunar RPOD and were conducted using an AMD Ryzen 9 7950X CPU and NVIDIA GeForce RTX 4090 GPU.Source: RESULTS AND ANALYSIS · p. 11

Cislunar Rendezvous Scenario

• The rendezvous uses a 6-hour segment of a 9:2 South NRHO, with lunar orbital radius varying from 2,782 km at the control-window start to 18,160 km at its end.The passage identifies this variation as spanning major dynamical regimes.Source: Cislunar Rendezvous Scenario · p. 12
• Figure 6 depicts a 10,000-trajectory sample from the full dataset of 100,000, which is used to train ART-TWIN for autoregressive warm-start generation.Source: Figure 6 · p. 13, Cislunar Rendezvous Scenario · p. 13

ART-TWIN vs. ART-0 Evaluation

• The ART-TWIN versus ART-0 comparison uses a randomly selected subdataset of 100 test trajectories.Source: ART-TWIN vs. ART-0 Evaluation · p. 13
• About 90% of ART-TWIN cases converge in exactly 3 SCP iterations, whereas over 60% of ART-0 cases reach the 20-iteration limit.ART-TWIN's breakpoint error is at the midpoint and is abated through SCP iterations; ART-0 has variable terminal-state error.Source: ART-TWIN vs. ART-0 Evaluation · pp. 13–14

ART-TWIN Warm-Starting Performance

• In a representative case, ART-TWIN’s warm-start had fewer and less severe passive-safety violations, and its SCP runtime was about 40% faster than CVX-SCP.The ART-TWIN initial guess was closer to the SCP solution; its midpoint discontinuity did not hinder rapid convergence.Source: ART-TWIN Warm-Starting Performance · pp. 14–15, Figure 9 · p. 16
• 1.59x median speedup over CVX was observed across 10,000 held-out samples, while suboptimality gaps were essentially the same.The average per-trajectory cost improvement was about 0.3–0.4 m/s, a 1–2% gain in total delta-V savings for this scenario.Source: ART-TWIN Warm-Starting Performance · pp. 15, 17, Figure 10 · p. 16
• ART-TWIN outperformed ART-0 by more than CVX on efficiency diagnostics, while ART-0’s high likelihood of failed convergence made it unable to match CVX’s optimality and efficiency.The conclusion characterizes forward-backward dual-arc generation as the upgrade that lets ART handle chaotic dynamical systems.Source: Figure 8 · p. 15, ART-TWIN Warm-Starting Performance · p. 17, CONCLUSION · p. 17

CONCLUSION

• The conclusion reports that ART-TWIN extends ART to cislunar relative motion beyond Earth orbit, with the evaluated rendezvous scenario beginning at perilune.The paper describes perilune as difficult for controllability because of rapidly varying dynamics.Source: CONCLUSION · p. 17
• Universal cislunar deployment remains future work: ART-TWIN needs more diverse rendezvous scenarios and chief orbits, and the study omits navigation uncertainties and several perturbations.Omitted perturbations include higher-resolution lunar gravity, solar third-body gravity, and solar radiation pressure.Source: CONCLUSION · p. 17
• The authors identify onboard capabilities for autonomous distributed space systems as relevant to cislunar RPOD and space domain awareness.Source: CONCLUSION · p. 17

Transformer Model Hyperparameters

• Table 3 presents the ART hyperparameter settings used in this work.The ART implementation uses Py-Torch and Hugging Face’s transformers library.Source: Transformer Model Hyperparameters · p. 18, Table 3 · p. 18

SCP Hyperparameters

• Table 4 presents the SCP hyperparameters used in the paper, with nomenclature corresponding to the original SCvx* algorithm.Source: SCP Hyperparameters · p. 18