The 2:1 resonant distant retrograde orbit (DRO), known for its long-term stability and global accessibility, holds strategic significance in current Earth–Moon space mission explorations. This paper conducts a comprehensive analysis of the problem of low-energy transferring into 2:1 DRO using the weak stability boundary (WSB) and lunar gravity assist (LGA) in the planar bi-circular restricted four-body problem (BCR4BP). The transfer process is categorized into three phases: the Earth–Moon transfer, Sun–Earth weak stability boundary transfer, and DRO low-energy capture. Addressing key questions, our study investigates: (1) Under what LGA conditions can the spacecraft reach the approximate area where the WSB region is situated? (2) How do trajectories, upon reaching the region where the WSB is located, return to the vicinity of 2:1 DRO, potentially facilitating low-energy DRO insertion? Our study involved a comprehensive analysis of the spacecraft's changes in Earth–Moon mechanical energy and Jacobi energy during the entire transfer process. This analysis yielded the energy and geometric conditions necessary for potential low-energy DRO insertion, effectively filtering out numerous impractical candidate trajectories and enhancing computational efficiency. In this paper, the geometric condition is referred to as the low-energy transfer gateway (LETG). Using the LEGT as the stitching interface, a significant number of feasible solutions were obtained effectively for bi-impulse DRO transfer trajectories through differential correction, some of which were previously undiscovered.

Distant Retrograde Orbit（DRO）is a family of large-scale, lunar-retrograde periodic orbits in the cislunar space. Due to its advantages of long-term stability and low-energy transfer, DRO has become a potential orbit for many cislunar space missions. Investigating close formation techniques on DRO is of great significance for cislunar on-orbit servicing. Considering navigation and execution errors, it is essential to study the uncertainty propagation of close relative motion on DRO and design stationkeeping strategies. The fundamental solution set of linearized relative motion on DRO obtained through the Floquet theory is introduced. Based on periodic solutions, analyses of sensitivity and safety uncertainty propagation of DRO formation flight are conducted using the Cauchy-Green tensor and unscented transformation, respectively. Based on these analyses and considering engineering constraints, it is found that keeping the maneuver frequency of 2 times per cycle and the maneuver locations at two perilunes is the near-optimal stationkeeping scheme. Following this, two stationkeeping algorithms are proposed based on the concepts of relative trajectory following and absolute phase bias. The simulation results show that both stationkeeping algorithms can ensure long-term safety and reasonable configuration of DRO close formation flight.

Midcourse correction design is key to space transfers in the cislunar space. Autonomous guidance has garnered significant attention for its promise to decrease the dependence on ground control systems. This study addresses the problem of midcourse corrections for Earth–Moon transfer orbits based on high-order state transition tensors (STTs). The scenarios considered are direct Earth–Moon transfers and low-energy transfers to lunar distant retrograde orbits (DROs), where the latter involve weak stability boundary (WSB) and lunar gravity assist (LGA) techniques. Semi-analytical formulas are provided for computing the trajectory correction maneuvers (TCMs) using high-order STTs derived using the differential algebraic method. Monte Carlo simulations are performed to evaluate the effectiveness of the proposed approach. Compared with existing explicit guidance algorithms, the STT-based approach is much cheaper computationally and features fewer final position errors. These results are promising for fast and efficient orbital autonomous correction guidance approaches in the cislunar space.

Distant Retrograde Orbits (DROs) in the Earth-Moon system have great potential to support varieties of missions due to the favorable stability and orbital positions. Thus, the close relative motion on DROs should be analyzed to design formations to assist or extend the DRO missions. However, as the reference DROs are obtained through numerical methods, the close relative motions on DROs are non-analytical, which severely limits the design of relative trajectories. In this paper, a novel approach is proposed to construct the analytical solution of bounded close relative motion on DROs. The linear dynamics of relative motion on DRO is established at first. The preliminary forms of the general solutions are obtained based on the Floquet theory. And the general solutions are classified as different modes depending on their periodic components. A new parameterization is applied to each mode, which allows us to explore the geometries of quasi-periodic modes in detail. In each mode, the solutions are integrated as a uniform expression and their periodic components are expanded as truncated Fourier series. In this way, the analytical bounded relative motion on DRO is obtained. Based on the analytical expression, the characteristics of different modes are comprehensively analyzed. The natural periodic mode is always located on the single side of the target spacecraft on DRO and is appropriate to be the parking orbits of the rendezvous and docking. On the basis of quasi-periodic modes, quasi-elliptical fly-around relative trajectories are designed with the assistance of only two impulses per period. The fly-around formation can support observations to targets on DRO from multiple viewing angles. And the fly-around formation is validated in a more practical ephemeris model.

Space swarms, enabled by the miniaturization of spacecraft, have the potential capability to lower costs, increase efficiencies, and broaden the horizons of space missions. The formation control problem of large-scale spacecraft swarms flying around an elliptic orbit is considered. The objective is to drive the entire formation to produce a specified spatial pattern. The relative motion between agents becomes complicated as the number of agents increases. Hence, a density-based method is adopted, which concerns the density evolution of the entire swarm instead of the trajectories of individuals. The density-based method manipulates the density evolution with Partial Differential Equations (PDEs). This density-based control in this work has two aspects, global pattern control of the whole swarm and local collision-avoidance between nearby agents. The global behavior of the swarm is driven via designing velocity fields. For each spacecraft, the Q-guidance steering law is adopted to track the desired velocity with accelerations in a distributed manner. However, the final stable velocity field is required to be zero in the classical density-based approach, which appears as an obstacle from the viewpoint of astrodynamics since the periodic relative motion is always time-varying. To solve this issue, a novel transformation is constructed based on the periodic solutions of Tschauner-Hempel (TH) equations. The relative motion in Cartesian coordinates is then transformed into a new coordinate system, which permits zero-velocity in a stable configuration. The local behavior of the swarm, such as achieving collision avoidance, is achieved via a carefully-designed local density estimation algorithm. Numerical simulations are provided to demonstrate the performance of this approach.

Laser Interferometer Space Antenna (LISA) is a project to detect and measure gravitational waves. The project has three spacecraft flying in a formation of near equilateral triangle in a heliocentric orbit trailing Earth. Many sources of perturbations cause the configuration to deviate from the nominal. This paper studies the formation design problem for a LISA-like mission by considering ephemeris-based dynamics. This type of mission is well-known for addressing several strict mission requirements under the realistic dynamics. The problem is formulated as optimizing multiple mission performance indices. It is observed that some indices are correlated with each other, whereas some indices have different sensitivities with respect to the semi-major axis. Therefore, the problem is transformed into a two-step cascade single-objective optimization, in which the optimal solution of the first optimization problem is fed to the second optimization as initial value. In addition, the major perturbing celestial bodies are picked up to make a simplified but accurate enough dynamics to speed up the optimization. Numerical examples verify the analysis and show the effectiveness of the optimization procedure. The influences of mission lifetime and spatial scales on the optimal solutions are also presented.

A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits. This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rotating frame with respect to the orbital elements. The resulted solution is expressed in terms of two different sets of orbital elements. The first one is the classical orbital elements and the second one is the nonsingular orbital elements. Among of them, however, the semi-latus rectum and true anomaly are used due to their generality, rather than the semi-major axis and mean anomaly that are used in most references. This specific selection for orbital elements yields a new solution that is universally applicable to elliptic, parabolic and hyperbolic orbits. It is shown that the new orbital element-based relative motion equations are equivalent to the Tschauner-Hempel equations. A linear map between the initial orbital element differences and the integration constants associated with the solution of the Tschauner-Hempel equations is constructed. Finally, the presented solution is validated through comparison with a high-fidelity numerical orbit propagator. The numerical results demonstrate that the new solution is computationally effective; and the result is able to match the accuracy that is required for linear propagation of spacecraft relative motion over a broad range of Keplerian orbits.