Shock-physics numerical codes are essential tools for describing the short but extreme fragmentation stage of the hypervelocity impact process on asteroids. However, accurately representing complex interior structures, surfaces, and contact mechanics in these events remains a significant challenge for traditional hydrocodes. This study introduces and validates an innovative yet underutilized technique, i.e., the Material Point Method (MPM), to simulate hyper-velocity impacts on asteroids. This approach offers new perspectives and solutions for capturing complex interfaces and handling the contact and boundary conditions in asteroid impact simulations. Our MPM implementation incorporates critical improvements to material models, including a pressure-dependent C1 continuous yield criterion with quantifiable plastic strain, and a resolution-independent Grady-Kipp fragmentation model, to capture the complex physics of geological materials under extreme conditions. The framework is rigorously validated against laboratory impact experiments and benchmarked with smoothed particle hydrodynamics (SPH) simulations, confirming its robustness and precision. Crucially, when applied to asteroid-scale collisions, our model successfully reproduces the formation of large, coherent fragments analogous to (433) Eros. This work establishes MPM as a validated and powerful extension to the planetary scientist’s toolkit, enabling the expansion of the parameter space and the treatment of complex contact and boundary conditions, which will enable more realistic simulations of asteroid evolution, family formation, and planetary defense scenarios.

Designing optimal trajectories for multi-flyby asteroid missions is scientifically critical but technically challenging due to nonlinear dynamics, intermediate constraints, and numerous local optima. This paper establishes a method that approaches global optimality for multi-flyby trajectory optimization under a given sequence. The original optimal control problem with interior-point equality constraints is transformed into a multistage decision formulation. This reformulation enables the direct application of dynamic programming in lower dimensions and follows Bellman's principle of optimality. Moreover, the method provides a quantifiable bound on global optimum errors introduced by discretization and approximation assumptions, thus ensuring a measure of confidence in the obtained solution. The method accommodates both impulsive and low-thrust maneuver schemes in rendezvous and flyby scenarios. Several computational techniques are introduced to enhance efficiency, including a specialized solution for bi-impulse cases and an adaptive step-refinement strategy. The proposed method is validated on three Global Trajectory Optimization Competition problems, showing improved fuel efficiency over the best-known solutions and demonstrating its generality and effectiveness in global trajectory optimization.

Binary asteroid systems are ubiquitous in the Solar System. Many of them originate from rotational breakup, where an asteroid's spin-up triggers mass shedding and subsequent satellite formation from a transient debris disk. While prolate satellites on compact orbits are expected in this scenario, recent space missions revealed remarkable diversity of binary configurations, such as the contact-binary satellite Selam on a wide orbit around (152830) Dinkinesh. Here we show that multiple episodes of mass shedding and the resulting multi-generation satellites provide a unified framework for these diverse configurations. We find that a pre-existing satellite can strongly influence subsequent satellite formation pathways through disk-satellite and inter-satellite interactions. This mechanism explains the dynamical histories of the Dinkinesh system and several triple systems. Our analysis indicates that about 44% of known binaries have configurations indicative of multi-satellite histories, suggesting that a significantly greater diversity of binary asteroid configurations remains to be revealed.

While most near-Earth asteroids (NEAs) are thought to originate from the main belt, recent discoveries have suggested the existence of a lunar-derived NEA population, such as the asteroids Kamo‘alewa and 2024 PT5. These objects may hold key clues to the dynamical evolution of NEAs and the recent impact history of the Earth–Moon system. However, the population, distribution, and dynamical characteristics of these lunar-origin asteroids (LOAs) remain poorly constrained. By combining the lunar ejecta production with N -body orbital simulations of the ejecta, we investigate their orbital evolution in the past millions of years and the current LOA population, revealing their significant potential for detection by future surveys. Specifically for the Vera C. Rubin Observatory’s upcoming Legacy Survey of Space and Time, we predict an average detection rate of about six LOAs (with D > 5 m) per year. Additionally, we find that the LOAs tend to approach from sunward and antisunward directions, with encounter velocities significantly lower than those of typical NEAs. These findings offer valuable insights in guiding targeted ground-based surveys and planetary defense efforts for LOAs in the future.

The reentry trajectory optimization problem with the generalized waypoint constraint is solved using the adaptive matchingcorrection strategy in this paper. Generally, the reentry trajectory optimization problems are complicated due to their highly nonlinear dynamics and complex path constraints. Particularly, this paper considers the generalized waypoint constraint and proposes the adaptive matching-correction strategy to handle it. Two steps are included in the proposed strategy: "matching step" and "correction step". The core idea is to find the edge that best matches the generalized waypoint constraint in the process of algorithm iteration, and then pull the edge into the constraint's zone. Compared with the traditional methods to cope with similar problems, the idea of multi-phase optimization is not needed, enhancing the algorithm's ability to tackle the abrupt appearance of the generalized waypoint. Extensive numerical simulations with comparisons are provided to demonstrate the superior performance of the proposed strategy. (c) 2025 Published by Elsevier B.V. on behalf of COSPAR.

Lunar gravity assists (LGAs) are widely utilized in mission designs to potentially reduce transfer costs, particularly in noncoplanar cases requiring inclination adjustments. The commonly used patched-conic model and Keplerian map have low accuracy in designing close LGAs within the Earth-Moon system. Propagating LGA trajectories in the circular restricted three-body problem (CR3BP) offers higher fidelity but requires numerical integration. Therefore, a fast and accurate approach to estimate the state variation of the spacecraft after an LGA is significant. In this work, a 3-D LGA mapping in the CR3BP is studied, where deep neural networks (DNNs) are utilized to predict the state variation, gravity assist radius, and gravity assist type within a single orbital period. The LGA mapping is formulated with specific input and output parameters. The dynamical characteristics and data features resulting from the large mass ratio of the Earth-Moon CR3BP are analyzed. In particular, a small gravity assist radius causes significant perturbations to the Keplerian orbit, leading to abrupt local variations in the mapping outputs and challenges in sampling and fitting. To address this, a sampling approach based on the physical interpretation of the mapping inputs is proposed to improve the dataset, and a prediction framework combining classification and regression DNNs is employed. Test results demonstrate the efficiency and accuracy of the proposed DNNs-based LGA mapping, with average relative errors of 1.17% for state variation and 0.45% for gravity assist radius, and a 99.90% classification accuracy. In addition, the error distribution and extrapolation capability are evaluated and analyzed.

TianQin, a space-based gravitational wave (GW) detector in geocentric orbit, maintains continuous laser links during science operations but requires biannual interruptions when the constellation enters Sun-pointing mode to protect optical payloads from direct sunlight exposure. These planned operational pauses are strategically repurposed for configuration maintenance to optimize subsequent observation cycles. This paper analyzes the trade-offs between the geometric stability and fuel consumption during active orbital control of the TianQin constellation. A segmented cooperative control strategy, based on virtual formation and virtual structure methods, is proposed to simplify the multi-objective optimization problem. The computational efficiency is further improved by fast estimating the optimization indices and by simplifying the dynamics. Due to the segmented reference orbits, this optimization method is applicable to general scenarios where the configuration naturally diverges. By analyzing both the diverging nominal and stable optimized configurations of TianQin, we quantitatively evaluate the benefits and costs of active configuration maintenance, revealing a non-uniformly conflicting relationship between the geometric stability and fuel consumption. The achievable geometric stability is determined in (c) 2025 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Terminal flight guidance problems are full of uncertainties. The deterministic trajectory optimization methods typically disregard uncertainties, which renders achieving precise target arrival impractical. Conventional covariance control approaches add directly uncertain terms to the right hand of the dynamic equations. The uncertainties' meaning is unclear. In addition, current approaches convert the stochastic optimal control problem into a semidefinite programming problem. Solving such problems is time-consuming even though the efficient solver is used. This paper proposes the second-order cone programming-based covariance control approach to mitigate the disadvantages above. Three types of uncertainties are considered to align closely with practical guidance scenarios. In particular, uncertainties involving spatial position correlations are included, which holds significant practical relevance. Moreover, a conservative approximation relationship for the second-order cone probability constraints is derived based on the exponential tail bounds of random variables. Finally, this paper converts the semidefinite programming problem into the second-order cone programming problem using the knowledge of the matrix norm, enhancing the real-time performance of the covariance control approaches. Numerical simulations are provided to verify the superior performance of the proposed approach.

The hopping rover, especially the internal-torque-actuated (hopping) rover, is currently recognized as one of the most suitable types for asteroid surface exploration. Although such rovers have been successfully demonstrated in the Hayabusa 2 mission, their dynamical and control problem of its hopping motion has not been studied enough. Consequently, this article first builds the rover's hopping dynamical model based on the polygonal contact model in considering the flywheel braking process. The hopping motion is then numerically simulated on two kinds of planes, including an ideally flat plane and a complex rocky terrain. Furthermore, the parametric studies of the rover's hopping process are implemented by considering the flywheel's nonconstant reaction torque during braking, the multicontact cases, the surface deformation, and the slip cases. Particularly, the phenomenon of multiple microcontacts between the rover and the asteroid surface before departure is discovered. The effect of the flywheel's braking process, contact stiffness, and friction coefficient on the hopping characteristics of the rover is investigated. Moreover, their intrinsic dynamical mechanisms are also illustrated. This study could provide a reference for future missions that require roving on the surface of a small celestial body.

Space-based gravitational wave detectors have attracted considerable attention for their ability to detect low frequency gravitational waves generated by the universe's most various sources. However, the high-precision detection requires extremely strict orbital insertion to maintain the configuration throughout the mission period. This paper presents an optimization method to correct orbital insertion errors using a low-frequency control strategy for long-duration, large-baseline spacecraft constellations. With active control, space-based gravitational wave detectors can mitigate initial orbit uncertainties. By presenting an evaluation algorithm and an estimating equation for fuel consumption and geometric indices, the segmented optimization method offers high computational efficiency. To adapt the control strategy to high-fidelity dynamics, an initial solution selection skill is proposed, and local refinements are employed. The application to the TianQin mission demonstrates the efficiency of the proposed method. The active control not only corrects the orbital insertion error but also reduces the geometric changes of the configuration by nearly 30% compared to an ideal deployment. The configuration maintenance strategy balances fuel efficiency and geometric stability, with a fuel consumption of only 421.57 m/s over the five-year mission period.

Introduction Recent advancements in computational power and model sophistication have revolutionized the study of hypervelocity impacts on asteroids, enabling the exploration of broader parameter spaces and more intricate asteroid structures [1, 2]. The outcome of such collisions, including crater formation and catastrophic disruption, is intricately linked to the internal structure of asteroids. However, the nature of this relationship remains an open question, necessitating further investigation. The Material Point Method (MPM), renowned for its ability to accurately track interfaces and resolve contact problems, emerges as a powerful tool for numerical simulations in this context [3]. By harnessing the capabilities of MPM, we aim to shed light on the correlation between an asteroid's structural composition and its response to hypervelocity impacts, ultimately contributing to a deeper understanding of the underlying physical processes. Method Developed from the particle-in-cell numerical method, the Material Point Method (MPM) combines Eulerian and Lagrangian descriptions, making it well-suited for handling boundary conditions and interface problems. By utilizing a dynamic background grid that only creates nodes with mapped material points for efficient computation, we further implement a contact algorithm in our MPM framework that could accurately calculate the real contact and friction force on each interface, and novelly extend the contact correction from two-object to multi-object scenarios. When the contact algorithm is activated, if a background grid node is shared by multiple objects, separate node information will be created for each object, along with the associated material point indices, as illustrated in Fig. 1. The execution of contact correction is triggered by assessing both the distance between objects and their relative motion based on non-penetration constraints. It is implemented by applying contact forces to eliminate penetration and enforce suitable friction conditions. Multi-body contact is resolved by solving a linear system of equations for the unknown contact forces between each pair of interacting objects. This algorithmic enhancement bolsters the capability of our MPM code to precisely simulate hypervelocity impacts on asteroids with intricate internal structures. Simulation and Discussion In this study, we conduct numerical simulations of artificial hypervelocity impacts on asteroids to quantitatively investigate the distinct dynamic responses of rubble-pile asteroids compared to monolithic, homogeneous ones. We employ material parameters of typical basaltic rocks in our simulations [4]. The impact scenarios and results for rubble-pile asteroids are illustrated in Fig. 2. Considering the crucial role of the catastrophic disruption threshold in understanding the evolutionary history of small celestial bodies and designing asteroid impact defense missions, as well as the availability of numerical values for specific asteroid sizes from previous research [5], we aim to quantify the influence of porosity and internal interfaces on impact disruption mechanisms by calculating the catastrophic disruption threshold for rubble-pile asteroids with various configurations. Our simulation results demonstrate the significant impedance effect of internal interfaces on shock wave propagation, leading to a substantially higher catastrophic disruption threshold for rubble-pile asteroids compared to monolithic asteroids of the same strength. This finding suggests that the presence of a rubble-pile structure enhances the resistance of asteroids to catastrophic disruption, which has important implications for understanding their fragmentation and reassembly processes throughout their evolutionary history. Furthermore, the quantitative analysis of the catastrophic disruption threshold for different rubble-pile configurations provides valuable insights into the role of porosity and internal structure in the impact response of asteroids. These findings contribute to the development of more accurate models for asteroid evolution and can inform the design of future asteroid impact defense strategies. Additionally, the results of this study have the potential to facilitate the calibration of asteroid ages based on their structural properties and impact history, enhancing our understanding of the chronology of asteroids in the solar system. Acknowledgment X.Y. and J.L. acknowledge support from the National Natural Science Foundation of China under Grant 12372047. X.Y. acknowledges support from the National Natural Science Foundation of China under Grant 62227901. P.M. acknowledges support from the French space agency CNES and from the French National Centre for Scientific Research (CNRS) through the exploratory research program of the Mission for Transversal and Interdisciplinary Initiatives. References [1] Raducan, S. D. et al. (2024). PSJ. 5(3). [2] Jutzi, M. et al. (2022) Nat. Commun. 13(1), 7134. [3] Yan, X. et al. (2023) ACM Conference. [4] Jutzi, M. (2015) P&SS. 107(1), 3–9. [5] Benz, W. et al. (1999) Icar. 142, 5–20.

This paper proposes an optimal, robust, and efficient guidance scheme for the perturbed minimum-time low-thrust transfer toward the geostationary orbit. The Earth's oblateness perturbation and shadow are taken into account. It is difficult for a Lyapunov-based or trajectory-tracking guidance method to possess multiple characteristics at the same time, including high guidance optimality, robustness, and onboard computational efficiency. In this work, a concise relationship between the minimum-time transfer problem with orbital averaging and its optimal solution is identified, which reveals that the five averaged initial costates that dominate the optimal thrust direction can be approximately determined by only four initial modified equinoctial orbit elements after a coordinate transformation. Based on this relationship, the optimal averaged trajectories constituting the training dataset are randomly generated around a nominal averaged trajectory. Five polynomial regression models are trained on the training dataset and are regarded as the costate estimators. In the transfer, the spacecraft can obtain the real-time approximate optimal thrust direction by combining the costate estimations provided by the estimators with the current state at any time. Moreover, all these computations onboard are analytical. The simulation results show that the proposed guidance scheme possesses extremely high guidance optimality, robustness, and onboard computational efficiency.

Many binary asteroid systems (‘Binary systems’) are believed to originate from Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) deformation; that is, the YORP spin-up of an asteroid triggers the rotational failure, and the material shed from the main body finally reaccumulates and forms its satellite(s). A binary system can form from a single YORP shedding event (‘SSh’), or from a sequence of such events (‘MultiSh’). In the latter one, a satellite formed in the first shedding event, thereby in the subsequent shedding events the evolution of shed material is influenced by the presence of this first-generation satellite. These different scenarios can result in various final configurations of binary systems. We aim to systematically investigate this relevance, and thereby obtain a ‘map’ of the various formation paths of binary systems.Our work primarily focuses on the MultiSh scenario, which is rarely concerned in previous works. We use 2 dimensionless parameters to characterize the perturbation effects of a satellite to the shed material. With semi-analytical investigation and discrete element method (DEM) simulations, we reconstructed the links between the character of an already-existed satellite and the final dynamical configuration of the binary system. Special attention is also paid to binary systems whose satellite itself is a contact-binary, such as the Dinkinesh system. We propose that this kind of binary systems can only form through some specific and interesting formation paths.This work has been supported by the National Natural Science Foundation of China grant No. 12372047.

Configuration stability is crucial for the detection accuracy of space-based gravitational wave (GW) detectors. This paper proposes a low-frequency hierarchical cooperative impulse control strategy and a high-fidelity orbit optimization method to achieve a formation of GW detectors with high detection performance. The control strategy employs a hierarchical structure to divide the configuration control into position and shape layers. In the upper layer, the position layer, a state transition tensor method is employed to analyze the sensitivity and feasible range of the formation center position on the configuration stability. In the lower layer, the shape layer, a novel consistency protocol cooperatively uses the states of all individuals in the formation to determine the control expectations. The constraint on formation position is relaxed to reduce the number of impulses, achieving a low-frequency impulse control. The optimization method, characterized by a careful selection of the optimization variables and a design of adaptive model continuation from a low-fidelity model to a high-fidelity model, is proposed to reduce the problem-solving difficulty. The proposed strategy and method are applied to the Laser Interferometer Space Antenna (LISA) mission, and the results demonstrate that the configuration stability is effectively enhanced with several low-frequency impulses.

Shape-based methods are commonly used in solving the low-thrust trajectory optimization problem. Designing a shape trajectory by maximizing a given performance index has been widely researched in recent years. The main contribution of this paper is to present a new pseudoequinoctial shaping method with parameter optimization to solve general low-thrust spacecraft rendezvous problems. The shaping method can solve the rendezvous problem with a large inclination difference between the initial and target orbits and greatly improve the performance index of the trajectory while meeting the thrust constraints. At the same time, the shape can provide a better feasible solution for solving the low-thrust trajectory optimization problem by direct methods. Another contribution is to introduce a general adjoint estimation for indirect methods. By linearizing the equation of motion near the proposed shape-based trajectory and constructing the Hamiltonian function, the initial guess of the adjoint variables can be obtained in a closed form. The general linearization formulations applicable to all coordinate systems are presented, and three numerical examples are given, in which the first example verifies the superiority of the proposed shape-based method, the second example tests the feasibility of the shape as the initial value for a direct method, and the last example compares the convergence rates of the initial adjoint variables generated by the proposed method under different coordinate systems for an indirect method.

Context. The Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect has been shown to effectively alter the rotational status of asteroids. The spin-up of the asteroid leads to surface instability and eventually triggers regolith failure, followed by landslide and mass shedding on the asteroid's surface. Aims. We explore the dynamics of the rotation-induced resurfacing and shedding, paying special attention to the dependence of post-shedding evolution on regolith mechanical properties, such as cohesion. Methods. We propose a qualitative semi-analytical model to explore the post-failure dynamics of a fast-rotating asteroid. We also consider the interaction between the surface mass rearrangement and the asteroid's spin status. We used our model to investigate the surface region where the failure occurs, as well as the total mass shed from the surface and the spin-down of the asteroid in this process. Results. Based on our model, all the possible avalanche events following a regolith failure can be classified into four basic types: resurfacing (ReS), shedding and resurfacing (S&ReS), shed and bound (S-Bound), and shedding and escaping (S-Escp). Their corresponding regions in the parameter space are illustrated in this work. Our results show that although the regolith cohesion is very small (less than or similar to 1-2 Pa), cohesion plays an important role in the onset of the avalanche. Moreover, our model qualitatively reconstructs the links between the regolith's properties and the dynamical fates of the shed material. The timescale of YORP-induced shedding events is also discussed in this work.

Introduction The successful DART mission has validated kinetic impact as the most effective method for deflecting or disrupting potentially hazardous asteroids [1]. The impact produced unexpectedly intense material ejection, expanding our understanding of asteroid response to hypervelocity collisions [2]. What insights might this violent fragmentation and ejecta reveal about Dimorphos's material strength and structural composition? Exploring these aspects before the upcoming Hera mission [3] can optimize mission design and interpretation.The artificial impact experiment conducted by the Hayabusa2 mission demonstrated the influence of geological structures on cratering events [4, 5]. Similarly, understanding how the material and structural composition of an asteroid affect kinetic impact efficiency is crucial for planetary defense. To capture the detailed structure of asteroids, here we employ the Material Point Method (MPM), which excels in handling contact and interface problems, for modeling and simulation, aiming to address these critical questions.Method The material point method (MPM) is a well-developed numerical technique in the mechanical industry, but remains underestimated potential in the field of planetary science. MPM discretizes the continuum body into Lagrangian material points and employs a rigidly attached Eulerian background grid at each time step, with the algorithmic flow illustrated in Fig. 1. By combining the strengths of both Lagrangian and Eulerian descriptions, MPM effectively eliminates numerical dissipation and expedites the search for neighbor points, resulting in an efficient, extendable, and robust shock-physics code. This feature makes MPM particularly well-suited for simulating complex geometries and contact interactions, which are common in planetary science applications. To accurately capture the dynamical behavior of materials under hypervelocity impact, a modified pressure-dependent and damage-dependent Lundborg smooth yield model is adopted for the strength model, coupled with a quantitative plasticity correction. The Tillotson equation of state (EOS) is utilized in conjunction with a sound speed modification. Besides, a resolution-independent Grady-Kipp damage model is employed to simulate the fracture process. In the future, we plan to further develop the p-alpha equation to characterize the pore compaction effect.These proposed algorithms and material models have undergone extensive benchmarking and validation, spanning a wide range of parameter spaces, demonstrating their capability to describe the potential dynamic behavior of small celestial body materials [6]. This comprehensive approach, combining state-of-the-art numerical methods and material models, paves the way for a deeper understanding of the dynamic properties of small celestial bodies and their constituent materials.Simulation and Result Constrained by computational resources, we designed simulation scenarios based on China's upcoming kinetic impact mission [7]. A cubic spacecraft (1.5 m in size, 600 kg in mass) impacts a spherical target body (35 m in diameter) at a velocity of 10 km/s. Given the unknown material and structure of the target, we investigated the dynamic responses of small bodies with various structures (monolithic or rubble pile), geometries (spherical or ellipsoidal), and material compositions (rocky or regolith, solid or porous). Except for the solid rocky material with a density of 2700 kg/m3, all other target bodies have equal mass and a bulk density of 1200 kg/m3. Classic basaltic material parameters were used for all materials, except for the regolith, which has lower strength. The ejecta, damage, and velocity distributions after 0.5 s of impact are visually presented in Fig. 2. We further analyzed the momentum enhancement factor and the mass and velocity distributions of the fragments for each case.Discussion and Conclusion The dynamic responses to hypervelocity impacts exhibit significant sensitivity to the target's material and structure, as evidenced by the ejecta intensity, ejection angles, fragment morphology, velocity, and angular velocity. Investigating the relationship between target structure and impact response is crucial for mission design and ejecta avoidance. Moreover, it helps unravel the physical properties of the target body from observational results. This study focused on perpendicular impacts and analyzed the results during the impact fragmentation phase. Future research may delve into the orbital dynamics of the fragments, evaluate the effectiveness of impact defense in mitigating Earth-threatening hazards, and explore the potential of oblique impacts to disrupt the target body through spin-up. These findings may also extend to impacts between asteroids and shed light on phenomena such as binary formation, which contributes to our understanding of the origin and evolution of small celestial bodies in the solar system.Acknowledgment X.Y. and J.L. acknowledge support from the National Natural Science Foundation of China under Grant 12372047. X.Y. acknowledges support from the National Natural Science Foundation of China under Grant 62227901. P.M. acknowledges support from the French space agency CNES and from the French National Centre for Scientific Research (CNRS) through the exploratory research program of the Mission for Transversal and Interdisciplinary Initiatives.References [1] Daly, R. T. et al. (2023) Nature 616(7957), 443–447. [2] Cheng, A. F. et al. (2023) Nature 616(7957), 457–460. [3] Michel, P. et al. (2022) Planet. Sci. J. 3(7), 160. [4] Arakawa, M. et al. (2020) Science 368(6486), 1–10. [5] Jutzi, M. et al. (2022) Nat. Commun. 13(1), 7134. [6] Yan, X. et al. (2023) ACM Conference. [7] Zou, Y. et al. (2024) J. Deep Space Explor. 11(2), 1-8.

This paper investigates the global optimization of multispacecraft successive rendezvous trajectories, which is divided here into three subproblems: target assignment, sequence optimization, and rendezvous time optimization. A method consisting of two novel algorithms is proposed to solve these subproblems. First, a multitree search framework is developed to assign multiple targets to each spacecraft and simultaneously optimize the rendezvous sequence for every single spacecraft. Specifically, a novel algorithm of local search combined with beam search is proposed. Second, this paper converts the rendezvous time optimization problem into a multistage decision problem. Based on a critical rendezvous-epoch-dependent characteristic found in this subproblem, the number of state variables is thereby reduced. A novel dual dynamic programming algorithm is proposed and combined with dynamic programming to solve for the globally optimal rendezvous epochs efficiently. Global optimality is guaranteed by Bellman’s principle of optimality, which is the first time in such a problem to our knowledge. The proposed method achieves state-of-the-art performance in several typical fuel-optimal scenarios of active debris removal. This open-sourced method is non-database-dependent and contains only one design stage, which is expected to be adopted in other successive rendezvous missions.

Space-based gravitational wave (GW) detection at low frequencies is of great scientific significance and has received extensive attention in recent years. This work designs and optimizes the low-energy transfer of the heliocentric formation of GW detectors, which starts from a geosynchronous transfer orbit and targets an Earth-like orbit. Based on the example of the Laser Interferometer Space Antenna (LISA), the transfer is first designed in two-body dynamical models and then refined in simplified high-fidelity dynamical models that only consider the major orbital perturbations evaluated here. The main contributions of this work are to present an adaptive model continuation technique and to exploit the lunar swingby technique to reduce the problem-solving difficulty and velocity increment of orbital transfer, respectively. The adaptive model continuation technique fully reveals the effect of perturbations and rapidly iterates the solutions to the simplified models. The simulation results show that the lunar swingby does reduce the energy needed to escape the Earth’s sphere of influence. It is found that the gravitation of the Earth–Moon system has a significant contribution to reducing the velocity increment. The solution of low-energy transfer in the simplified models is that the duration is 360.6615 days and the total velocity increment is 0.8468 km/s.

The irregular shapes of small bodies usually lead to non-uniform distributions of mass, which makes dynamic behaviors in the vicinities of small bodies different to that of planets. This study proposes shape entropy (SE) as an index that compares the shapes of small bodies and spheres to describe the shape of a small body. The results of derivation and calculation of SE in two-dimensional and three-dimensional cases show that: SE is independent of the size of geometric figures but depends on the shape of the figures; the SE difference between a geometric figure and a circle or a sphere, which is the limit of SE value, reflects the difference between this figure and a circle or a sphere. Therefore, the description of shapes of small bodies, such as near-spherical, ellipsoid, and elongated, can be quantitatively described via a continuous index. Combining SE and the original inertia index, describing the shape of small bodies, can define the shapes of small bodies and provide a reasonably simple metric to describe a complex shape that is applicable to generalized discussion and analysis rather than highly detailed work on a specific, unique, polyhedral model.