Low-thrust Trajectory Design Research Articles

Orbital Averaging Advancement for Low-Thrust Trajectory Design

Orbital Averaging Advancement for Low-Thrust Trajectory Design

Shape-based method for low-thrust transfers between periodic orbits in cislunar space

Low-thrust trajectory design in the context of the three-body problem, particularly in the vicinity of liberation points, is addressed in this paper with a focus on efficiency and accuracy. The Fourier series, previously successful in approximating near-coplanar low-thrust trajectories within two-body dynamics, is applied to the challenging landscape of the three-body problem, marked by instability and nonlinearity. This paper introduces a novel approach for constructing an initial guess trajectory, that combines coast arcs from periodic orbits and intermediate trajectory arcs from other natural dynamical structures. The resulting initial guess trajectory is instrumental in generating efficient initial estimates for the coefficients of the Fourier series. The study encompasses the analysis of homoclinic and heteroclinic transfers in both two-dimensional and three-dimensional scenarios. The approximated trajectories derived from this methodology serve as invaluable starting points for high-fidelity optimization solvers, offering promise for enhancing the precision and efficiency of low-thrust transfers in cislunar space.

Adaptive Estimation of Distribution Algorithms for Low-Thrust Trajectory Optimization

A direct adaptive scheme is presented as an alternative approach for minimum-fuel low-thrust trajectory design in non-coplanar orbit transfers using fitness landscape analysis (FLA). The spacecraft dynamics is modeled with respect to modified equinoctial elements by considering orbital perturbations. Taking into account the timings of thrust arcs, the discretization nodes for thrust profile, and the solution of multi-impulse orbit transfer, a constrained continuous optimization problem is formed for low-thrust orbital maneuvers. An adaptive method within the framework of the estimation of distribution algorithms (EDA) is proposed, which aims at conserving the feasibility of the solutions within the search process. Several problem identifiers for low-thrust trajectory optimization are introduced, and the complexity of the solution domain is analyzed by evaluating the landscape feature of the search space via FLA. Two adaptive operators are proposed, which control the search process based on the need for exploration and exploitation of the search domain to achieve optimal transfers. The adaptive operators are implemented in the presented EDA and several perturbed and nonperturbed orbit transfer problems are solved. The results confirm the effectiveness and reliability of the proposed approach in finding optimal low-thrust transfer trajectories.

Low-thrust trajectory design for near-Earth asteroid supply delivery and resource transportation mission using high-power solar electric propulsion

This paper presents the low-thrust (LT) trajectory design for near-Earth asteroid (NEA) supply delivery and resource transportation mission using high-power Solar Electric Propulsion (SEP). NEA low-thrust roundtrip accessibility is investigated through a numerical approach and new knowledge of the mass transportation capability of SEP systems in NEA missions is provided. A number of powerful deep neural networks (DNNs) were trained and optimized for 60 kW, 150 kW, 300 kW and 500 kW SEP-based LT trajectory design. DNNs are used as the surrogate of the conventional optimization process. A new low thrust trajectory roundtrip opportunity search algorithm is developed, which integrates the DNN models and allows an efficient roundtrip search process. Compared to the conventional approach, the DNN-based search algorithm reduces the search time by 99.96%. The SEP mass transportation capabilities were numerically studied. It was found the 60 kW SEP can deliver 5 tons of supply to up to 40% of NEAs. Considering refueling on NEAs, the accessible NEAs can be further increased by 16.2%. It is suggested that the 150 kW, 300 kW, 500 kW SEP to be used to transport 10–30 tons, 30–70 tons, 70–120 tons resources from NEAs respectively. These mass ranges ensure about 4,000 or more targets to be accessible. In all the case studies, the roundtrip accessible boundaries were constructed using polynomial regression. These boundaries were numerically obtained, which could be useful for mission designers to rapidly determine the LT roundtrip accessibility of newly found NEAs.

Approximations for Secular Variation Maxima of Classical Orbital Elements under Low Thrust

The reachability assessment of low-thrust spacecraft is of great significance for orbital transfer, because it can give a priori criteria for the challenging low-thrust trajectory design and optimization. This paper proposes an approximation method to obtain the variation maximum of each orbital element. Specifically, two steps organize the contribution of this study. First, combined with functional approximations, a set of analytical expressions for the variation maxima of orbital elements over one orbital revolution are derived. Second, the secular approximations for the variation maxima of the inclination and the right ascension of the ascending node are derived and expressed explicitly. An iterative algorithm is given to obtain the secular variation maxima of the other orbital elements the orbital elements other than the inclination and right ascension of the ascending node. Numerical simulations for approximating the variation maxima and a preliminary application in estimation of the velocity increment are given to demonstrate the efficiency and accuracy of the proposed method. Compared with the indirect method used alone for low-thrust trajectory optimization, the computation burden of the proposed method is reduced by over five orders of magnitude, and the computational accuracy is still high.

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.

Low-Thrust Trajectory Design from Lunar Rideshare to Venus Capture

Small spacecraft are likely to be launched as part of a rideshare mission rather than on a dedicated launch. These spacecraft typically use low-thrust propulsion systems as they have lower fuel mass requirements. However, without a dedicated launch, the lower thrust can make it difficult to reach interplanetary targets. Further mission design difficulties arise as the optimizers used to search for such trajectories often require a good initial guess, which can be difficult to find. This work patches together a perturbed Sims–Flanagan transcription in three different regimes, each with a different central body, to perform a preliminary search for trajectories from a lunar rideshare to a weak capture around Venus with little input required from the mission designer. Example trajectories of such a mission are presented. The ability of this mission architecture to arrive as close to Venus’s upper atmosphere as possible is explored.

Gravity-Assist Low-Thrust Inter-System Trajectory Design with Manifold Captures

Gravity-Assist Low-Thrust Inter-System Trajectory Design with Manifold Captures

Low-Thrust Trajectory Design with Successive Convex Optimization for Libration Point Orbits

In space missions, numerical techniques for minimizing the fuel consumption of spacecraft using low-thrust propulsion are desirable. Among various nonlinear optimization methods, convex optimization has been attracting attention because it allows optimal solutions to emerge robustly and requires short computation times. In particular, trajectory design in the three-body problem near the Lagrange points involves instability and nonlinearity. Hence, this study considers the application of convex optimization to trajectory design with low-thrust propulsion in cislunar space and verifies that this technique performs well, even for sensitive and highly nonlinear dynamics. Specifically, the convex optimization scheme is applied in the transfer from a halo orbit to a near-rectilinear halo orbit where both are periodic as defined in the circular restricted three-body problem. As a further step, the transfer problem is transitioned to an ephemeris model. In addition, extensive investigations of the dependence on various parameters used in convex optimization are conducted, and a trajectory corrections maneuver method is constructed combined with the convex optimization process. This investigation provides valuable insight into the convex optimization technique in the three-body problem and facilitates the estimation of low-thrust trajectory designs for complex space missions.

Mapped adjoint control transformation method for low-thrust trajectory design

Variational approach to optimal control theory converts trajectory optimization problems into two- or multiple-point boundary-value problems, which consist of costates (i.e., Lagrange multipliers associated with the states). Estimating missing values of the non-intuitive costates is an important step in solving the resulting boundary-value problems. By leveraging costate vector mapping theorem, we extend the method of Adjoint Control Transformation (ACT), called Mapped ACT (MACT), to alternative sets of coordinates/elements for solving low-thrust trajectory optimization problems. In particular, extension of the ACT method to the set of modified equinoctial elements and an orbital element set based on the specific angular momentum and eccentricity vectors (h-e) is demonstrated. The computational and robustness efficiency of the MACT method is compared against the traditionally used random initialization of costates by solving 1) interplanetary rendezvous maneuvers and 2) an Earth-centered, orbit-raising problem with and without the inclusion of J2 perturbation. For the considered problems, numerical results indicate two to three times improvement in the percent of convergence of the resulting boundary-value problems when the MACT method is used compared to the random initialization method. Results also indicate that the h-e set is also a contender and suitable choice for solving low-thrust trajectory optimization problems.

Low-Thrust Gravity-Assist Trajectory Design Using Optimal Multimode Propulsion Models

The complexities in using indirect optimization methods get compounded for practical co-optimization problems in the presence of continuous and discrete design variables. In this paper, realistic multimode electric propulsion systems are incorporated within the formulation of gravity-assist, low-thrust trajectory design problems. Electric thrusters operate over a large set of discrete operation modes, and each mode is characterized by specific values for power, thrust, specific impulse, and mass flow rate. Unlike the traditional methods that approximate the aggregate behavior of electric thrusters’ performance (using polynomials fits), a novel construct is proposed to incorporate optimal discrete operating modes within indirect optimization. The capability of the tool and the efficacy of the proposed methodology are demonstrated by solving a multiyear, fuel-optimal trajectory problem from Earth to asteroid Psyche via a Mars gravity-assist maneuver and using a Hall-effect SPT-140 thruster with 21 operating modes. Comparisons with the traditional polynomial-based thruster modeling are presented.

Transit and capture in the planar three-body problem leveraging low-thrust invariant manifolds

Path planning in the circular restricted 3-body problem (CR3BP) is frequently guided by the forbidden regions and manifold arcs. However, when low thrust is employed to modify the spacecraft trajectory, these dynamical structures pulsate with the varying Hamiltonian value. In a combined CR3BP, low-thrust (CR3BP + LT) model, an additional low-thrust Hamiltonian is available that remains constant along low-thrust arcs given suitable assumptions. Accordingly, the analogous low-thrust forbidden regions and manifolds remain static and are useful guides for low-thrust trajectory design. Strategies leveraging these structures and other insights from the CR3BP + LT are explored to construct transit and capture itineraries.

Exploiting manifolds of [formula omitted] halo orbits for end-to-end Earth–Moon low-thrust trajectory design

In this work, end-to-end low-thrust transfers from a GTO orbit to a low-altitude lunar orbit by exploiting the manifolds of a chosen Earth–Moon L1 halo orbit has been studied. The practicality of piece-wise, minimum-time transfers that exploit halo orbit manifolds is demonstrated, which offers more flexibility to meet mission objectives. It is known that the structure of the manifolds varies substantially due to the presence of the Sun and its contribution has to be considered to obtain more realistic trajectories. To incorporate Sun’s perturbation, we study (1) manifolds’ behavior within a Bi-Circular Problem (BCP) dynamics and (2) Sun’s impact on the previously converged trajectories obtained using the standard Circular Restricted Three-Body Problem (CR3BP). Comparisons of the resulting trajectories using the CR3BP and BCP are presented.

Low-thrust trajectory design in low-energy regimes using variational equations

This paper proposes a novel description of the equations of motion for low-thrust trajectory design in the presence of a third-body perturbation. The framework is formulated using Gauss’ Variational Equations (GVE) with two distinct accelerations: the one produced by the electric engine and the disturbing term of the third-body effect, which is computed using the disturbing potential of the previously studied Keplerian Map. The presented GVE third-body (GVE-3B) framework allows for a simple and intuitive description of the low-thrust optimisation problem. It is accurate until very close to the sphere of influence of the perturbing body, and thus can be used to target trajectories in low-energy regimes. Together with the framework, this paper develops a methodology to generate low-energy first-guess solutions for low-thrust trajectories. Both the methodology and the framework are showcased in the design of two distinct missions: a rendezvous with asteroid 2017 SV19 during its next Earth encounter, after departing from the unstable invariant manifold of the L2 point in the Sun-Earth system, and the capture of asteroid 2018 AV2 to a stable invariant manifold of the same point.

Initial design of low-thrust trajectories based on the Bezier curve-based shaping approach

This paper presents a method to use the Bezier curve to rapidly generate three-dimensional low-thrust trajectories, which can provide a suitable initial approximation to be used for more accurate trajectory optimal control tools. Two missions, from Earth to Mars and the asteroid Dionysus, are considered to evaluate the performance of the method. In order to verify the advantages of this method, it is compared with the finite Fourier series method. Numerical results show that the Bezier method can get better performance index in shorter computation time compared with the finite Fourier series method. The applicability of the solution obtained by Bezier method is evaluated by introducing the obtained solution into the Gauss pseudospectral method as an initial guess. The simulation results show that the Bezier method can rapidly generate a very suitable three-dimensional initial trajectory for the optimal solver. This is very important for rapid evaluation of the feasibility of a large number of low-thrust flight schemes in the preliminary mission design stage.

Trajectory Design Leveraging Low-Thrust, Multi-Body Equilibria and their Manifolds

A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously specify the spacecraft position and velocity vectors, as well as the thrust history. To mitigate this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP) are investigated as candidate solutions to seed initial low-thrust trajectory designs. The addition of a low-thrust force to the CR3BP modifies the locations and stability of the equilibria, offering novel geometries for mission applications. Transfers between these novel equilibria are constructed by leveraging the associated stable and unstable manifolds and insights from the low-thrust CR3BP.

Tube Stochastic Optimal Control for Nonlinear Constrained Trajectory Optimization Problems

Recent low-thrust space missions have highlighted the importance of designing trajectories that are robust against uncertainties. In its complete form, this process is formulated as a nonlinear constrained stochastic optimal control problem. This problem is among the most complex in control theory, and no practically applicable method to low-thrust trajectory optimization problems has been proposed to date. This paper presents a new algorithm to solve stochastic optimal control problems with nonlinear systems and constraints. The proposed algorithm uses the unscented transform to convert a stochastic optimal control problem into a deterministic problem, which is then solved by trajectory optimization methods such as differential dynamic programming. Two numerical examples, one of which applies the proposed method to low-thrust trajectory design, illustrate that it automatically introduces margins that improve robustness. Finally, Monte Carlo simulations are used to evaluate the robustness and optimality of the solution.

Low-Thrust Trajectory Design Using Finite Fourier Series Approximation of Pseudoequinoctial Elements

A new low-thrust trajectory design method is proposed that is based on the finite Fourier series method with pseudoequinoctial elements rather than the more common cylindrical coordinate components. The bijection relation between the elements and control variables is ensured by introducing an additional equality constraint derived from the angular momentum conservation. The guidance law and on-line control variables are obtained by applying inverse dynamics and the framework of inverse simulation technology, respectively. The pseudoequinoctial finite Fourier series method has the advantages of both the Fourier series and the perturbation analysis methods. For two-body problems, three cases were studied: the Earth to Mars, 1989ML, and Tempel-1 missions. Regarding the design of a rendezvous trajectory with a large inclination angle and a high eccentricity rate, this method yields a broader range of feasible results than the traditional Fourier series method. The circular restricted three-body problem was solved for the first time using the pseudoequinoctial finite Fourier series method combined with the patched conics method. The low-thrust Earth-Moon transfer was analyzed, and the results show that this method improves window analysis efficiency and guarantees precision of the initial geocentric trajectory for the low-thrust transfer.

Systematic low-thrust trajectory design to Mars based on a full Ephemeris modeling

One of the fundamental tasks in space mission design is to choose a set of inter-disciplinary mission-critical parameters that are used for both sizing spacecraft sub-systems and designing optimal trajectories. Trajectory design and sub-system sizing are tightly coupled tasks and mission designers are interested in algorithms that not only improve fidelity of the underlying models, but also facilitate comprehensive trade-off studies using dependable algorithms. This paper presents a systematic-design/computationally-efficient framework that makes use of a recently developed hybrid optimization method, which is a fusion between homotopic approach and particle swarm optimization to perform a robust homotopic approach. A salient feature of this framework is the flexibility in altering the fidelity of the dynamical models to beyond the conventional two-body model by including perturbations due to: (1) other planets of the Solar System, (2) solar radiation pressure, and (3) the oblateness effects of the Earth. Moreover, a comprehensive study on the impact of using different types of thrusters, different hyperbolic excess velocity values, and different launch opportunities is conducted. Extensive numerical simulations are performed for a heliocentric rendezvous mission from Earth to Mars and the results are compared against those in the literature.

Fast preliminary design of low-thrust trajectories for multi-asteroid exploration

Multiple-asteroid exploration with low-thrust propulsion requires the design of transfer trajectory and the selection of the visiting sequence, as well as the estimation of the propellant budget. To this end, this paper presents a method to generate transfer trajectories and visiting sequences rapidly using finite Fourier series (FFS) and Monte-Carlo Tree Search (MCTS). The FFS method can generate the transfer trajectory rapidly, which can provide suitable initial approximations, leading to more accurate trajectory optimizations. This study adopts the MCTS algorithm for asteroid sequence selection in the multi-asteroid exploration. By comparing with the traversal algorithm, the greedy algorithm and the tree search algorithm with the trimming strategy, the numerical results show that the MCTS can be used to obtain a quasi-optimal sequence with higher probability and less computation time. Consequently, a method combining FFS and MCTS can rapidly acquire the quasi-optimal visiting sequences with a large probability and the suitable initial trajectory for a multi-asteroid exploration mission. This is very important for the rapid feasibility assessment of hundreds of flight scenarios at the preliminary mission design stage.