Transfer Trajectory Research Articles

Multi-objective low-thrust spacecraft trajectory design using reachability analysis

One of the fundamental problems in spacecraft trajectory design is finding the optimal transfer trajectory that minimizes the propellant consumption and transfer time simultaneously. We formulate this as a multi-objective optimal control (MOC) problem that involves optimizing over the initial or final state, subject to state constraints. Drawing on recent developments in reachability analysis subject to state constraints, we show that the proposed MOC problem can be stated as an optimization problem subject to a constraint that involves the sub-level set of the viscosity solution of a quasi-variational inequality. We then generalize this approach to account for more general optimal control problems in Bolza form. We relate these problems to the Pareto front of the developed multi-objective programs. The proposed approach is demonstrated on two low-thrust orbital transfer problems around a rotating asteroid.

Optimal Circle-to-Ellipse Orbit Transfer for Sun-Facing E-Sail

The transfer between two coplanar Keplerian orbits of a spacecraft with a continuous-thrust propulsion system is a classical problem of astrodynamics, in which a numerical procedure is usually employed to find the transfer trajectory that optimizes (i.e., maximizes or minimizes) a given performance index such as, for example, the delivered payload mass, the propellant mass, the total flight time, or a suitable combination of them. In the last decade, this class of problem has been thoroughly analyzed in the context of heliocentric mission scenarios of a spacecraft equipped with an Electric Solar Wind Sail as primary propulsion system. The aim of this paper is to further extend the existing related literature by analyzing the optimal transfer of an Electric Solar Wind Sail-based spacecraft with a Sun-facing attitude, a particular configuration in which the sail nominal plane is perpendicular to the Sun-spacecraft (i.e., radial) direction, so that the propulsion system is able to produce its maximum propulsive acceleration magnitude. The problem consists in transferring the spacecraft, which initially traces a heliocentric circular orbit, into an elliptic coplanar orbit of given eccentricity with a minimum-time trajectory. Using a classical indirect approach for trajectory optimization, the paper shows that a simplified version of the optimal control problem can be obtained by enforcing the typical transfer constraints. The numerical simulations show that the proposed approach is able to quantify the transfer performance in a parametric and general form, with a simple and efficient algorithm.

Trajectory optimization of space vehicle in rendezvous proximity operation with evolutionary feasibility conserving techniques

In this paper, a direct approach is developed for discovering optimal transfer trajectories of close-range rendezvous of satellites considering disturbances in elliptical orbits. The control vector representing the inputs is parameterized via different interpolation methods, and an Estimation of Distribution Algorithm (EDA) that implements mixtures of probability models is presented. To satisfy the terminal conditions, which are represented as non-linear inequality constraints, several feasibility conserving mechanisms associated with learning and sampling methods of the EDAs are proposed, which guarantee the feasibility of the explored solutions. They include a particular implementation of a clustering algorithm, outlier detection, and several heuristic mapping methods. The combination of the proposed operators guides the optimization process in achieving the optimal solution by surfing the regions of the search domain associated with feasible solutions. Numerical simulations confirm that space transfer trajectories with minimum-fuel consumption for the chaser spacecraft can be obtained with terminal condition satisfaction in rendezvous proximity operation.

Phasing with near rectilinear Halo orbits: Design and comparison

In the next future, space agencies are planning to return to the Moon. The objective is to assemble an orbiting space station, called Gateway, on a Near Rectilinear Halo Orbit around the Moon as a base for future Moon and deep space missions. Within this framework, multiple side missions will be planned to sustain the Gateway (Artemis mission). The proposed work is thought in framework of the preliminary design of future cargo missions, in particular on the design of an efficient phasing trajectory, under the Circular Restricted Three body problem hypotheses, to bring a cargo vehicle from the end of the Earth-Moon transfer to the beginning of the proximity operations such as rendezvous and docking with the space station. The work aims covering the lack of literature in phasing trajectories with the NRHO by proposing three different strategies to connect the Earth-Moon transfer trajectory with the proximity operations. The three strategies are classified based on the choice of the parking orbits or the choice of the manifolds. Two strategies use butterfly and Halo orbits to park the vehicle before transferring to the target orbit. The third strategy, instead, uses manifolds to allow a direct phasing. In the paper, the three innovative strategies are designed and compare in a specific scenario.

Optimal On-Orbit Inspection of Satellite Formation

In a formation-flying mission where multiple spacecraft must cooperate and maintain a prescribed relative separation, the early detection of possible anomalies is a primary requirement. This is possible, for example, by employing an inspector spacecraft whose aim is to monitor the condition of the formation members with an on-orbit inspection. This paper analyzes a rest-to-rest multiple-impulse transfer that the inspector spacecraft must accomplish to visit all of the formation members. The problem is studied using the linearized Hill–Clohessy–Wiltshire equations and is solved in an optimal framework by minimizing the total velocity variation along the transfer trajectory. The solution algorithm implements a two-step procedure that combines differential evolution algorithms and Nelder–Mead simplex method-based routines. A case study is thoroughly investigated where a formation of six satellites covers a circular orbit of altitude 300km over Earth. The proposed algorithm could efficiently find a solution and with reduced computational times.

Improved Monte Carlo Tree Search-based approach to low-thrust multiple gravity-assist trajectory design

In the preliminary design of low-thrust trajectories with multiple gravity-assist phases, the selection of the celestial bodies to be considered and the sequence of optimal maneuvers have a significant impact on the overall mission performance. This aspect makes the preliminary transfer trajectory analysis a very complex task. This Short Communication presents a novel method to design heliocentric multiple gravity-assist transfer trajectories for a spacecraft with a low-thrust propulsion system, where the total velocity variation is minimized. In this context, a Bézier shape-based method is used to describe the generic arc of a propelled trajectory, while an extension of the Monte Carlo Tree Search algorithm is used for a rapid analysis of the gravity-assist maneuver sequence. In particular, the proposed approach effectively solves the problem of selecting a suitable balance parameter in the Monte Carlo Tree Search-based routines, which makes the method especially useful in a preliminary mission phase. The effectiveness of the proposed approach is shown by simulating an interplanetary transfer towards the outer regions of the Solar System.

Transfer between the planar Lyapunov orbits around the Earth–Moon L2 point using low-thrust engine

We present a study of the low-thrust transfer problem near the secondary body in the Earth–Moon circular restricted three-body problem (CRTBP). Based on the Pontryagin maximum principle, the time-optimal, energy-optimal and fuel-optimal transfers between the planar Lyapunov orbits around the L2 point of the Earth–Moon system are constructed as a two-point boundary value problem (TPBVP). A criterion for identifying the best trajectory is also described. The Lyapunov orbit structure is used to deal with the potential difficulties of initial guess. As a result, the final transfer trajectory follows a multi-revolution structure, which takes a long time of flight. The homoclinic connection configuration is investigated by introducing the relevant invariant manifolds, allowing for faster transfer and less fuel consumption in the same case. It is designed a linear time-varying model predictive control (MPC) algorithm to perform trajectory tracking in the presence of initial orbit error and deviations in engine operation. Numerical simulations show the efficiency and practicability of the optimal transfer trajectory design strategy and trajectory tracking controller proposed in this paper for future lunar exploration missions.

Four-body cislunar quasi-periodic orbits and their application to ballistic lunar transfer design

Examining the properties of quasi-periodic orbits provides insight into the Sun-perturbed environment in cislunar space. In this investigation, quasi-periodic trajectories and their properties are explored in the Sun-Earth-Moon four-body problem. Computation and the stability characteristics of families of invariant tori are detailed. Furthermore, this investigation offers a framework for construction of ballistic lunar transfer trajectories in the four-body problem. The framework leverages manifold trajectories to supply a set of initial conditions for construction of periapsis Poincaré maps. Periapsis maps reduce the dimensionality of the space and illuminate solutions of interest as a basis to produce feasible families of transfers. Through a continuation process, families of ballistic transfers and families of transfers that include powered lunar flybys are constructed. Ultimately, these solutions supply an initial guess for transition to the Sun-Earth-Moon ephemeris model.

Bayesian Optimization of crewed lunar free return abort trajectory

Maneuvering to a free return trajectory is a vital abort means for crewed lunar missions. A novel method of solving the fuel optimization of free return abort trajectory for Earth-Moon transfer by two-stage transformation on optimization model and Bayesian optimization is documented. Two new design variables—perilune altitude and transfer time from earth to perilune, instead of the original injection velocity vector, are introduced to transform the general model into a two-dimensional model. Each feasible abort trajectory is determined by a numerical method including initial estimate and exact solution. Then the characteristics of the relationship between the new variables and the fuel objective of impulse maneuver are applied to further transform the model into a one-dimensional model. Based on the Bayesian optimization method, fuel optimization is successfully and efficiently implemented for the free return abort trajectory at multiple abort moments during the Earth-Moon transfer. Furthermore, the influence of the initial Earth-Moon phase on the result of fuel optimization is presented. Numerical simulations are conducted to demonstrate the efficacy of the proposed method.

DNN estimation of low-thrust transfer time: Focusing on fast transfers in multi-asteroid rendezvous missions

In the preliminary design phases of low-thrust multi-asteroid missions, there are so many alternative transfer trajectories that it is impossible to simultaneously optimize them with high fidelity. Therefore, a fast and accurate method to evaluate the cost of low-thrust transfer is significant. In this work, a low-thrust time-optimal multi-asteroid rendezvous mission is studied, where deep neural networks (DNNs) are utilized to estimate the minimum time of flight (TOF) of a single-leg transfer between every two asteroids. By restricting the initial states during the dataset generation and constructing new loss functions, the DNN prediction accuracy of fast transfers, i.e., the transfers with short TOFs, is greatly improved while the overall accuracy is not discounted. Furthermore, beam search is modified to optimize the sequences of multi-asteroid missions. The solution sequences retained after the extension of each level in beam search are optimized by an indirect method to eliminate the impact of estimation error on subsequent searches. Finally, the DNNs are well trained and employed in the modified beam search to find the optimal sequence. A time-optimal multi-rendezvous example of 10 main-belt asteroids is investigated. The results demonstrate that the DNNs can accurately estimate the minimum TOF and that the proposed methods can achieve better sequences than those previously known.

Integrated Design of Moon-to-Earth Transfer Trajectory Considering Re-Entry Constraints

The exploration of the Moon has always been a hot topic. The determination of the Moon-to-Earth transfer opportunity and the design of the precise transfer trajectory play important roles in manned Moon exploration missions. It is still a difficult problem to determine the Moon-to-Earth return opportunity for accurate atmospheric re-entry and landing, through which the actual return trajectory can be easily obtained later. This paper proposes an efficient integrated design method for Moon-to-Earth window searching and precise trajectory optimization considering the constraints of Earth re-entry and landing. First, an analytical geometry-based method is proposed to determine the state of the re-entry point according to the landing field and re-entry constraints to ensure accurate landing. Next, the transfer window is determined with the perilune heights, which are acquired by inversely integrating the re-entry state under the simplified dynamics as criterion. Then, the precise Moon-to-Earth trajectory is quickly obtained by a three-impulse correction. Simulations show the accuracy and efficiency of the proposed method compared with methods such as the patched-conic method and provide an explicit reference for future Moon exploration missions.

Decoding Mission Design Problem for NTP Systems for Outer Planet Robotic Missions

This paper discusses the current challenges of exploration of outer planets and proposes a nuclear thermal propulsion (NTP) system for future deep space exploration missions. The mission design problem with respect to the NTP system is presented where it is proposed that NTP-powered missions need to integrate the requirements and constraints of mission objective, spacecraft design, NTP system design, and launch vehicle limits into a self-consistent model. The paper presents a conceptual NTP-powered rendezvous mission to Neptune that uses a single high-performance–class commercial launch vehicle to deliver over 2 mT of useful payload in a direct transfer trajectory with total trip time being under 16 years.

Optimal transfers from Moon to [formula omitted] halo orbit of the Earth-Moon system

In this paper, optimal solutions are investigated for a transfer from a parking orbit around the Moon to a halo orbit around L2 of the Earth-Moon system. The transfers are executed by applying a single maneuver and exploiting the stable invariant manifold of the hyperbolic parking solution at arrival. In this regard, an optimization problem is proposed where both the orbital characteristics of a parking solution around the Moon (its Keplerian elements) and the characteristics of a transfer trajectory (guided by the stable manifold of the arrival Halo orbit) are considered as variables. The problem involved in the single maneuver transfer is solved using a nonlinear programming method (NLP), which aims to minimize the cost of ΔV within the framework of the Earth-Moon system using the circular restricted three-body problem. The feasibility of this kind of transfer for a Cubesat is shown in this paper through results with low ΔV combined with suitable times of flight.

Spatial–temporal evolution of ESV and its response to land use change in the Yellow River Basin, China

The value of ecosystem services, as well as their temporal and spatial characteristics, can be used to help areas develop focused and localized sustainable ecological management plans. Thus, this study conducted in the Yellow River Basin (YRB) of China, analyzed the ecosystem service value (ESV) and its spatial–temporal variation characteristics. This study used the equivalent factor and geospatial exploration methods, introduced the elasticity coefficient, and explored the response of ESV change to land-use change, based on the land use cover data from 1990 to 2020. The results showed that from 1990 to 2020, YRB ecosystem service value showed an overall increasing trend, mainly because the ecological construction project increased forest and grasslands in this region. In the past 30 years, spatial characteristics of ESV in YRB was relatively stable. The high-value areas were mainly distributed in the upper Yellow River Basin, while the low-value areas were mainly distributed in the lower Yellow River Basin, as the cold and hot spots were reduced. The ESV barycenter coordinates showed the direction of the transfer trajectory, which is first to southwest, northeast, and then to southwest. From 2000 to 2010, YRB land-use change had greater impact on ESV. Since 2010, the disturbance of ecosystem services by land-use change has decreased. Consequently, the elastic index of the upstream and Loess Plateau regions were significantly higher than that of other regions, and the impact of land-use change on ecosystem services was more obvious, due to improved large-scale ecological construction projects implementation. Conclusively, this study recommends the use of comprehensive spatial–temporal assessment of ESV for sustainable development and ecological protection in the YRB.

Design and analysis of the transfer trajectory for switching landing site in emergency during the return flight of crewed lunar exploration mission

Based on the need for switching the landing site in an emergency during the return flight of a crewed lunar exploration mission, an emergency transfer trajectory is designed and analysed. First, the problem of generating a single transfer trajectory for switching the landing site in an emergency is equivalently converted to solving a Lambert problem. Second, a calculation method based on homotopic continuation is proposed to obtain the high-fidelity solution of the Lambert problem in the complex gravitational field of the Earth-Moon system. Based on this calculation method, the design algorithm of an emergency transfer trajectory is put forward, and an improved differential evolution algorithm is used for optimization. The results of simulation examples indicate that the calculation method of the Lambert problem based on homotopic continuation has good convergence, and also verify the effectiveness and feasibility of the trajectory design algorithm proposed in this paper. Finally, extensive simulation calculations are carried out by adopting this method, and the impulsive manoeuvre characteristic of the emergency transfer trajectory is discussed. The research conclusions can provide a reference for the design of an emergency rescue scheme during the return flight of a crewed lunar exploration mission in the future.

GTOC-11: Results from the University of Alabama

In this work, Team ASRL’s solution approach for the 11th Global Trajectory Optimization Competition (GTOC-11) is described. This problem tasked the competing teams with constructing a futuristic Dyson Ring utilizing materials acquired from the asteroid belt. In total, 10 motherships would depart from Earth in the year 2121 and visit as many asteroids as possible. After visiting each asteroid, a low-thrust propulsion module would transfer the material down to the desired final Dyson stations. The final approach utilized a deterministic tree search that involved alternating between fixed time of flight Lambert searches and solutions to the full-fidelity optimal control problem. Once a single tour had been constructed, transfer trajectories were computed for each asteroid to as many of the building stations as possible. After computing a pool of thousands of these completed legs, a bin packing algorithm was used to determine the highest scoring combination of 10 solutions. This search process was implemented in Python using the soon-to-be-released trajectory optimization tool, ASSET. Ultimately, the team finished 5th with a score of 5525.38.

Energy analysis and optimizations of two-impulse Earth-Moon transfers in the Sun-Earth-Moon system

This paper investigates the energy change of the two-impulse Earth-Moon trajectory in the Sun-Earth-Moon bicircular model analytically. An analytic expression is first derived to describe the change of the Jacobi value of the two-impulse Earth-Moon trajectory in the Sun-Earth-Moon system. Then the analytical results of the total cost of the Earth-Moon transfer can be obtained by solving the equations related to the change in the Jacobi value of the transfer trajectory. Numerical results show that the analytic expressions can provide good descriptions of the energy changes of the two-impulse Earth-Moon transfer. Moreover, it indicates that the total cost mainly depends on a critical angle, termed the perilune angle. Accordingly, we explore the correlation of the total cost and the perilune angle. Finally, optimizations of two-impulse Earth-Moon transfers are further carried out.

GTOC11: Methods and results from the team of Harbin Institute of Technology

This paper presents the methods and the results submitted by the team of Harbin Institute of Technology in the 11th Global Trajectory Optimization Competition (GTOC11). The mission of GTOC11 is to design the Mother Ship trajectories to execute asteroid flybys selected from over 83000 candidates, and then transfer these asteroids to 12 stations to build the “Dyson Ring” while maximizing the total performance index. The building mission is completed by five steps: “Dyson Ring” orbit selection, database creation, Mother Ship trajectory design, building station construction strategy, and asteroid transfer trajectory optimization. The database creation is mainly used to fast estimate the transfer time for the asteroid low-thrust trajectory, and it plays an essential role in the whole mission. Finally, the building of the “Dyson Ring” is realized within the given period. The final index of the submitted result is 5208.3463, which ranked seventh in GTOC11.

Trajectory Analysis and Optimization of Hesperides Mission

A challenging problem from a technological viewpoint is to send a spacecraft at a distance of about 600 au from the Sun, comparable with that of the Sun’s gravitational focus (that is, the general relativistic focusing of light rays, whose minimum solar distance is obtained when the light rays are assumed to graze the Sun’s surface), and reach it in a time interval on the order of a human working lifetime. A suitably oriented telescope at that distance would be theoretically able to observe exoplanets tens of light years far away and possibly to discover new life forms. The transfer trajectory of this mission is rather complex and requires a close selection of a suitable propulsion system, which must be able to provide the probe with the necessary energy to cruise at a velocity greater than 10 au/year. An effective outline of the these concepts is given by the Hesperides mission, originally proposed by Matloff in 2014. An interesting aspect of this mission proposal is the combination of a nuclear electric propulsion system and a classical solar sail that are jointly exploited to reach the necessary solar system escape velocity. However, the trajectory analysis reported by Matloff is very simplified and is essentially concentrated on a rough estimate of the time required by the spacecraft to reach a distance of 600au. Starting from the Hesperides baseline mission proposal, including the vehicle mass distribution, the aim of this work is to give a detailed mission analysis in an optimal framework. In particular, the spacecraft minimum time trajectory is calculated with indirect methods and a parametric analysis is made to highlight the impact of the main design parameters on the total flight time. The simulations show a substantial reduction of the mission time when compared with the original study.

Low-Thrust Orbit Dynamics and Periodic Trajectories in the Earth–Moon System

This study employs the circular restricted three-body problem (CR3BP) as the dynamical framework, for the purpose of investigating low-thrust orbit dynamics in the Earth–Moon system. First, the effect of low thrust on some dynamical structures that exist in the CR3BP is analyzed. Low-thrust capture and escape dynamics in the proximity of the Moon is investigated for preliminary mission analysis. Then, low-thrust periodic orbits—with potential practical application—are detected. To do this, the theorem of mirror trajectories, proven 6 decades ago, is extended to low-thrust trajectories. This represents the theoretical premise for the definition and use of a numerical search methodology based on modified Poincaré maps. This approach leads to identifying several low-thrust periodic orbits in the Earth–Moon system that are infeasible if only unpowered paths are considered. Two possible applications of low-thrust periodic orbits are described: (a) cycling transfer trajectories that connect Earth and Moon continuously, and (b) non-Keplerian periodic paths about the Moon, with potential use as operational orbits for satellite constellations.