This paper comprehensively studies the self-organized deployment of a spacecraft cluster on a smooth irregular closed curve for cooperative on-orbit observation of a target spacecraft. The expected configuration curve is constructed by a set of control points through cubic B-spline interpolation, and it is closed and C2 continuous on the entire curve including its endpoints. As a necessary step for designing the control law, an efficient iterative algorithm for calculating the maneuvering spacecraft's nearest point on the curve is developed using Newton downhill method. Then the double-layer feedback controllers are designed, in which the inner controller makes the temporary target positions approximately uniformly converge to the curve, and the outer controller guides the actual positions to the temporary target positions. In this way, the spacecraft cluster can be accurately deployed on the desired curvilinear formation. The stability of the closed-loop system with the double-layer controller is also proved. Finally, a boat-shaped target configuration curve for comprehensively cooperatively observing a spherical target spacecraft is simulated and analyzed. The detailed results demonstrate that the designed self-organized control law can successfully achieve the desired formation flying with low control magnitude, acceptable accuracy, and reduce the influence of single spacecraft's failure to the final configuration's uniformity and the observation tasks.

This paper investigates the dynamics modeling and structural optimization of an asymmetric two-stage torsion pendulum designed for drag-free testing in the Taiji mission. This torsion pendulum serves as a critical experimental apparatus for ground-based verification of drag-free control technology in space gravitational wave detection, addressing limitations in dynamic stability and parameter applicability found in traditional testbeds. Using the Lagrangian dynamics method, the equations of motion relative to inertial space are derived and simplified into a linearized dynamics model under the assumption of small-amplitude oscillations. A state-space approach is further employed to analyze the system’s free oscillation behavior, with equilibrium stability rigorously assessed through eigenvalue analysis. Compared to existing approaches, the proposed model significantly enhances computational efficiency and systematically reveals the influence of key structural parameters on system stability. The study identifies critical parameter ranges essential for ensuring system stability, with optimization results demonstrating that proper design and adjustment of structural parameters can substantially improve system robustness and performance. Numerical simulations validate the accuracy of the proposed models and methods, with the optimization scheme showing clear superiority in enhancing system performance and simplifying experimental design. This work establishes a rigorous theoretical framework for ground-based verification of drag-free control technology. It not only effectively addresses bottlenecks in traditional testbed designs but also offers innovative guidance for the development of experimental systems in the Taiji mission.

The classification of periodic relative motion in spacecraft dynamics is well-established, yet non-periodic motion remains underexplored. This paper addresses this gap by proposing a comprehensive classification model based on relative orbital dynamics. The elements of relative motion are defined, leveraging the characteristics of relative orbital dynamics to enable a detailed analysis of motion patterns. The analysis is then conducted from two perspectives: the geometrical configuration of the trajectory and the positional relationship between the trajectory and the spacecraft. A total of 19 motion styles are classified, including 14 non-periodic types, 3 periodic types, and 2 degenerate fixed-point cases, with corresponding dynamic conditions identified. Two representative applications demonstrate the practical utility of the classification: rapid target approach using flyby styles and hovering observation using droplet styles. Numerical simulations validate the accuracy of the classification and its applicability to real-world orbital scenarios. These findings provide a valuable framework for on-orbit services, such as formation flying, rendezvous, and orbital maintenance.

This paper conducts a comprehensive study on the multi-constrained two-on-one impulsive orbital pursuit–evasion game (OPEG). Firstly, considering constraints such as maneuverability, fuel reserves, and mission duration, a mathematical game model for the two-on-one impulsive OPEG is established, which transforms the two-on-one impulsive OPEG, where cooperation and competition coexist, into a multi-constrained three-party optimization problem suitable for solving with multi-agent deep reinforcement learning. Then, an intelligent solution method for cooperative game strategies based on the Multi-Agent Deep Deterministic Policy Gradient (MADDPG) algorithm is proposed. In the reward function design section, a reward function based on fixed-time triggering is introduced to address the information loss problem caused by long impulse intervals. To ensure good convergence of the algorithm and guide the spacecraft to learn effective cooperative strategies during training, an immediate reward function is designed, incorporating outcome rewards, guidance rewards, and cooperative rewards. Numerical simulations validate the feasibility and effectiveness of the proposed method. To further analyze the cooperative mechanisms learned by the spacecraft during algorithm training, a comparative experiment with the one-on-one impulsive OPEG is designed. The experimental results demonstrate that the two pursuers in the two-on-one impulsive OPEG not only develop various strategies such as “pre-emptive interception”, “pincer interception”, and “trailing pursuit” during training, but also improve mission success rates and reduce mission durations through coordinated efforts. Additionally, this paper reveals the impact of the relative initial state distribution between the two pursuing spacecraft and the evading spacecraft on the effectiveness of cooperation.

This paper presents a novel machine learning approach designed to efficiently solve the classical two-body problem. The inherent structure of the two-body problem involves the integration of a system of second-order nonlinear ordinary differential equations. Conventional numerical integration techniques that rely on small computation steps result in a prolonged computational time. Moreover, calculus has limitations in resolving the two-body problem, inevitably converging towards an unresolved Kepler equation of a transcendental nature. To address this issue, we integrate the conventional analytical solution based on true anomaly with a deep neural network representation of the Kepler equation. This results in a highly accurate closed-form solution that is solely dependent on time, which is termed a learning-based solution to the two-body problem. To enhance the precision, a correction module based on Halley iteration is introduced, which substantially improves the final solution in terms of precision and computational cost. Compared to state-of-the-art methods such as the piecewise Padé approximation, Adomian decomposition method, and modified Mikkola’s method, our approach achieves a computational speedup of several thousand to tens of thousands, while maintaining accuracy in large-scale orbit propagation scenarios. Empirical validation under simulated conditions underscores its effectiveness and potential value for long-term orbit determination.

This paper investigates the application of the Nash equilibrium solution method within 2-versus-1 impulsive orbital pursuit–evasion (P-E) scenarios, involving 2 pursuers and an evader. Through the integration of game theory and coordinated strategies between the pursuers, the initial 2-pursuer 1-evader game ((P₁, P₂) - E) is transformed into a composite 1-pursuer 1-evader game (P₂ - (P₁ - E)). To address the core challenge of the P-E game, we utilize the MinMax bilateral optimization algorithm to determine optimal strategies in each game iteration, ensuring fairness and equal opportunities for all involved parties. Within the composite P-E framework, the second pursuer (P₂) assumes responsibility for executing a coordinated pursuit strategy, including the evaluation and tracking of the anticipated outcome ofP₁ − E. Subsequently, the evader formulates an optimal counterplay by reverse engineering the potential role assignments and strategies of the pursuers. In order to explore the intricate aspects of these scenarios, our study harnesses Monte Carlo statistical methods, offering insights into critical factors such as initial positions, impulse intervals, and magnitudes of delta-V within orbital settings, all of which substantially influence game outcomes. Ultimately, this research not only advances our understanding of multiagent orbital P-E dynamics but also establishes a foundation for more informed and effective strategic planning in practical space missions. It aims to ensure mission success and responsible resource allocation in the domain of space exploration.

This study proposes a novel method for intention recognition of space noncooperative targets using large language models (LLMs). Traditional methods rely on motion data to assess orbital motion intentions but cannot infer operation and task intentions from multi-source information like images. LLMs, with their logical reasoning capabilities, can address this limitation. The intentions are categorized into 3 types and 23 subtypes based on multi-source information and their characteristics: motion intentions (e.g., “hovering”, “flyby”, and “rendezvous”), operation intentions (e.g., “docking”, “refueling”, and “repair”), and task intentions (e.g., “detection”, “surveillance”, and “attack”). The proposed method constructs LLMs for spacecraft intention recognition, involving prompt classification, template design, and test sample generation. The use of prompt tuning V2 (P-tuning V2) and low-rank adaptation (LoRA) fine-tuning enhances the models’ performance. A dataset of 50,688 nominal samples and 8,448 perturbed samples was created through computer simulation based on expert knowledge, focusing on intention recognition of approaching targets in space station on-orbit operation and surveillance scenarios. The models were tested under 3 prompt conditions: basic, instruction, and chain-of-thought (CoT). The performance of 6 models (ChatGLM2-6B and ChatGLM3-6B base and fine-tuned models) was analyzed. Notably, the LoRA fine-tuned ChatGLM3-6B model on instruction prompts achieved 99.9% accuracy, with improved robustness compared to the base model. This work presents a pioneering application of LLMs for spacecraft intention recognition, offering valuable insights for future research and applications.

This paper addresses the blocking problem against a satellite attempting a geostationary or geosynchronous transfer. Calculation and optimization methods based on reachable domain and Lambert interception are designed for the case of in-plane and nonplanar transfer orbits. Numerical validations against Hohmann transfer orbits were conducted to substantiate the proposed methods. Results indicate that, when blocking in-plane geostationary transfers, the blocker’s optimal stand-by orbit is the circular geostationary orbit. The mission-capable rates of in-plane elliptical blocking orbits, if properly arranged, resemble those of circular blocking orbits. When blocking nonplanar geosynchronous transfers, if only one blocker is involved, the optimal blocking orbit is circular. However, when multiple satellites collaborate on the same nonplanar blockade, the optimal blocking orbit is similar to a quasi-zenith orbit while the required capabilities for each blocker can be considerably reduced. Results indicate that geostationary or geosynchronous blocking can be achieved without occupying the precious geostationary orbit. Subsequently, the paper analyzes the impacts of non-Hohmann transfer orbits, inclined geostationary transfers, nonimpulsive maneuvers, and reaction times. The analysis pertaining to specific scenarios illustrates the interpretation and the definition used for orbital blockades; the engineering value of the computed results is revealed, providing insights for future research on orbital games.

The problem of maneuvering for a servicing spacecraft (inspector) to inspect a non-cooperative spacecraft (evader) in cislunar space is investigated in this paper. The evader, which may be a malfunctioning or uncontrolled satellite, introduces uncertainties due to its potential maneuvering capabilities. To address this challenge, the scenario is modeled as a special orbital game, incorporating the unique complexities of the cislunar environment. A variable-duration, turn-based inspection and anti-inspection game model is designed. The model defines both players’ rules, constraints, and victory conditions, providing a framework for non-cooperative inspection. Strategies for both players are developed and validated based on their dynamical properties. The inspector’s strategy integrates two-body Lambert transfers with shooting methods, while the evader’s strategy aims to maximize the inspector’s fuel consumption. Simulation results show that the evader’s optimal strategy involves deliberate fluctuations in its lunar periapsis altitude, with the inspector’s required ΔV up to eight times greater than the evader’s. The impact of game constraints is evaluated, and the effectiveness of deploying the inspector in low lunar orbit is compared with the inspector at the Earth-Moon Lagrange point L₁. The strengths and weaknesses of both are shown. These findings provide valuable insights for future orbital servicing and orbital games.

This paper explores multi-impulse trajectory design for approaching Geostationary Earth Orbit (GEO) targets, emphasizing the attainment of a 2:1 elliptical relative orbit for close inspection. The intricate optimization challenge includes dynamic and terminal constraints, as well as total mission duration limitations. Critical variables like transfer duration, impulse count, timing, and entry point are optimized using genetic algorithms to minimize velocity increment requirements. These optimal values are influenced by uncertain terminal states constrained by the 2:1 circling orbit. To address these complexities, the paper leverages a linear state transition matrix from relative orbital dynamics to formulate a multi-impulse optimization model. Additionally, a differential correction-based iterative approach mitigates nonlinear dynamics’ impact on terminal rendezvous errors. Through theoretical analysis and simulations, the study elucidates variations in optimization criteria, particularly in GEO missions with multiple impulses, and identifies optimal entry points for different impulse counts. The proposed models and methodologies offer theoretical insights for future GEO orbit approaching missions and potential applications in various space maneuver tasks.