Low-thrust Trajectories Research Articles

Event-Driven Network Model for Space Mission Optimization with High-Thrust and Low-Thrust Spacecraft

Numerous high-thrust and low-thrust space propulsion technologies have been developed in the recent years with the goal of expanding space exploration capabilities; however, designing and optimizing a multimission campaign with both high-thrust and low-thrust propulsion options are challenging due to the coupling between logistics mission design and trajectory evaluation. Specifically, this computational burden arises because the deliverable mass fraction (i.e., final-to-initial mass ratio) and time of flight for low-thrust trajectories can vary with the payload mass; thus, these trajectory metrics cannot be evaluated separately from the campaign-level mission design. To tackle this challenge, this paper develops a novel event-driven space logistics network optimization approach using mixed-integer linear programming for space campaign design. An example case of optimally designing a cislunar propellant supply chain to support multiple lunar surface access missions is used to demonstrate this new space logistics framework. The results are compared with an existing stochastic combinatorial formulation developed for incorporating low-thrust propulsion into space logistics design; this new approach provides superior results in terms of cost as well as utilization of the vehicle fleet. The event-driven space logistics network optimization method developed in this paper can trade off cost, time, and technology in an automated manner to optimally design space mission campaigns.

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.

Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

This paper presents a fail-safe polynomial-based method for determining the coast-terminating zero (CTZ) of fuel-optimal, time-fixed, low-thrust trajectories. The CTZ of the fuel-optimal bang-bang control is decided by the zeros of the switching function (SF). The state and costate differential equations along the coast arcs of the optimal trajectory have closed-form solutions; thus efficiently determining the CTZ is important. Existing methods suffer potential failures or are limited to time-free problems. The polynomial-based method consists of three steps. First, the SF is simplified by a coordinate transformation that eliminates the three orientation-related orbital elements. Then, the SF is rewritten as a combination of power and trigonometric functions of the argument eccentric anomaly, and the coefficients in the SF are simply evaluated using the states and costates at the periapsis. Finally, the SF is described as a polynomial of the eccentric anomaly by replacing the trigonometric functions with polynomials. The roots of the polynomial can provide an accurate estimation of the CTZ. Based on a Monte Carlo simulation, SF curves whose greatest possible numbers of zeros are $1,2,\ldots,7$ are listed. For each curve, the CTZ is solved by both the presented polynomial-based method and the existing sampling-searching method. The performance comparison demonstrates that the polynomial-based method provides superior reliability and efficiency.

Multi-revolution low-thrust trajectory optimization using symplectic methods

Optimization of low-thrust trajectories that involve a larger number of orbit revolutions is considered a challenging problem. This paper describes a high-precision symplectic method and optimization techniques to solve the minimum-energy low-thrust multi-revolution orbit transfer problem. First, the optimal orbit transfer problem is posed as a constrained nonlinear optimal control problem. Then, the constrained nonlinear optimal control problem is converted into an equivalent linear quadratic form near a reference solution. The reference solution is updated iteratively by solving a sequence of linear-quadratic optimal control sub-problems, until convergence. Each sub-problem is solved via a symplectic method in discrete form. To facilitate the convergence of the algorithm, the spacecraft dynamics are expressed via modified equinoctial elements. Interpolating the non-singular equinoctial orbital elements and the spacecraft mass between the initial point and end point is proven beneficial to accelerate the convergence process. Numerical examples reveal that the proposed method displays high accuracy and efficiency.

Direct multiple shooting transcription with polynomial algebra for optimal control problems under uncertainty

This paper proposes a novel approach to the solution of optimal control problems under uncertainty (OCPUUs). OCPUUs are first cast in a general formulation that allows the treatment of uncertainties of different nature, and then solved with a new direct transcription method that combines multiple shooting with generalised polynomial algebra to model and propagate extended sets. The continuity conditions on extended sets at the boundary of two adjacent segments are directly satisfied by a bounding approach. The Intrusive Polynomial Algebra aNd Multiple shooting Approach (IPANeMA) developed in this work can handle optimal control problems under a wide range of uncertainty models, including nonparametric, epistemic, and imprecise probability ones. In this paper, the approach is applied to the design of a robust low-thrust trajectory to a Near-Earth Object with uncertain initial conditions. It is shown that the new method provides more robust and reliable trajectories than the solution of an analogous deterministic optimal control problem.

Optimization of Variable-Specific-Impulse Gravity-Assist Trajectories

This paper presents a series of optimization methods for finding fuel-optimal gravity-assist trajectories that use the practical solar electric propulsion with variable specific impulse. The fuel-optimal low-thrust gravity-assist trajectories are first solved using specific impulse as a control variable. An optimization method combining the particle swarm optimization and the indirect method is proposed to optimize the variable-specific-impulse gravity-assist low-thrust trajectories. Additionally, the modified logarithmic homotopy function is applied to serve as a gateway in variable-specific-impulse case. The algorithm for optimizing multiple-gravity-assist trajectories is also presented in this paper. Moreover, the experimental data of NASA’s Evolutionary Xenon Thruster (NEXT) are employed in numerical simulations. Main-belt asteroids missions are given to substantiate the feasibility and efficiency of the proposed methods.

High-fidelity trajectory design to flyby near-Earth asteroids using CubeSats

Fast development of CubeSat technology now enables the first interplanetary missions. The potential application of CubeSats to flyby near-Earth asteroids is explored in this paper in consideration of CubeSats' limited propulsive capabilities and systems constraints. Low-energy asteroid flyby trajectories are designed assuming a CubeSat is initially parked around to the Sun-Earth Lagrange points. High-impulse and low-thrust trajectories with realistic thrusting models are computed first in the Circular Restricted Three-Body Problem (CR3BP), and then in a high-fidelity ephemeris model. Analysis in the ephemeris model is used to confirm that trajectories computed in the CR3BP model also exist in a more realistic dynamical model, and to verify the validity of the results obtained in CR3BP analysis. A catalogue of asteroid flyby opportunities between years 2019 and 2030 is provided, with 80 m/s of available ΔV and departure from halo orbits around the first and second Sun-Earth Lagrange points (of similar size to those typically used by scientific missions). Results show that the CR3BP model can serve as an effective tool to identify reachable asteroids and can provide an initial estimation of the ΔV cost in the ephemeris model (with ±15 m/s accuracy). An impulsive maneuver model can also provide an accurate estimation of the ΔV requirement for a CubeSat equipped with a high-impulse thruster (with 4 m/s accuracy), even if its thrust magnitude is small and requires duty cycling; low-thrust ΔV requirements, however, may differ significantly from the impulsive results (±15 m/s).

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.

Deep networks as approximators of optimal low-thrust and multi-impulse cost in multitarget missions

In the design of multitarget interplanetary missions, there are always many options available, making it often impractical to optimize in detail each transfer trajectory in a preliminary search phase. Fast and accurate estimation methods for optimal transfers are thus of great value. In this paper, deep feed-forward neural networks are employed to estimate optimal transfer costs to three types of optimization problems: the transfer time of time-optimal low-thrust transfers, the fuel consumption of fuel-optimal low-thrust transfers, and the total Δv of minimum-Δv J2-perturbed multi-impulse transfers. To generate the training data, both considered categories of low-thrust trajectories are optimized using the indirect method, and the J2-perturbed multi-impulse trajectories are optimized using J2 homotopy and particle swarm optimization. The hyper-parameters of deep networks are determined by grid search, random search, and the tree-structured Parzen estimators approach. Results show that deep networks are capable of estimating the final mass or time of optimal transfers with a mean relative error of less than 0.5% for low-thrust transfers and less than 4% for multi-impulse transfers. Our results are also compared with other off-the-shelf machine-learning algorithms and the generalization capabilities of the developed deep networks for predicting cases well outside the training data are investigated. Applications in multitarget mission design are also investigated.

Application of the Angular Independent Variable and Its Regularizing Transformation in the Problems of Optimizing Low-Thrust Trajectories

The expediency of using the true longitude or the associated angle as an independent variable in the problems of optimizing multi-revolution, low-thrust trajectories of spacecraft is considered. The auxiliary longitude is introduced, which is used further as a new independent variable instead of time. The advantage of using the auxiliary longitude as an independent variable in the problem of optimizing the trajectories with a fixed angular range and optimum transfer time is demonstrated. The problems of optimizing the trajectories of spacecraft with an ideally controlled, limited-power engine and a limited-thrust engine with a constant exhaust velocity are considered. The possibility of improving the convergence of the methods of solving the problems of optimizing the trajectories with a limited-thrust engine is considered. This improvement is achieved by joint utilization of the smoothing approximation of a relay thrust-switching function and the regularizing transformation of an independent variable, which extends the switching function values in the vicinity of thrust switching instants. Based on the results, the method for solving the problems of optimizing multi-revolution transfers is proposed; the numerical examples of its application are presented.

Low-Energy Sub-Optimal Low-Thrust Trajectories to Libration Points and Halo-Orbits

The paper considers the problem of calculating direct and low-energy, low-thrust trajectories to the libration points of the Earth–Moon system and to halo-orbits. A method for solving the problem is proposed. It consists of calculating stable manifolds of libration points or halo-orbits and calculating a low-thrust trajectory from an initial circular Earth orbit to the given point of this manifold using sub-optimal feedback control. With a fixed final mass of a spacecraft, the last stage of calculations is reduced to solving the Cauchy problem. The numerical examples are given for the calculation of direct and low-energy trajectories to libration points and to halo-orbits and the optimization of entry points to stable manifolds for low-energy trajectories.

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.

Combined Chemical–Electric Propulsion for a Stand-Alone Mars CubeSat

Stand-alone interplanetary CubeSats require primary propulsion systems for orbit maneuvering and precise trajectory control. The current work focuses on the design and performance characterization of the combined chemical–electric propulsion systems that shall enable a stand-alone 16U CubeSat mission on a hybrid high-thrust–low-thrust trajectory from a supersynchronous geostationary transfer orbit to a circular orbit about Mars. The high-thrust chemical propulsion is used to escape Earth and to initiate stabilization at Mars. The low-thrust electric propulsion is used in heliocentric transfer, ballistic capture, and circularization. For chemical propulsion, design and performance characteristics of a monopropellant thruster and feed system using ADN-based FLP-106 propellant are presented. For electric propulsion, a performance model of an iodine-propelled inductively coupled miniature radiofrequency ion thruster is implemented to calculate the variation of thrust, specific impulse, and efficiency with input power. A power-constrained low-thrust trajectory optimization using the thruster performance model is pursued to calculate the transfer time, , and the required propellant mass for fuel-optimal and time-optimal transfers. Overall, the combined chemical–electric systems yield a feasible propulsion solution for stand-alone CubeSat missions to Mars that balances propellant mass and transfer time.

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.

Design and optimization of low thrust transfer trajectory for engineering constraints

Multimode electric propulsion system is widely used in deep space exploration. Its thrust is output in stages with the change of input power, which also means that continuous power changes lead to discontinuous thrust changes. In the design and optimization of low-thrust trajectory, the discontinuity of thrust is often neglected and regarded as a continuous parameter. The optimal design of low-thrust transfer trajectory constrained by multimode electric propulsion is studied, and the influence of thrust grading characteristics on the trajectory is analyzed in this paper. A specific optimization strategy is proposed, including three main steps: model discretization based on direct method, fast generation of initial value based on particle swarm optimization (PSO), and optimization based on sequential quadratic programming (SQP). The dynamic model and optimization model of low-thrust transfer trajectory is established. Then, the optimization model is discretized based on the direct method and the original optimal control problem is transformed into a parameter optimization problem in the form of nonlinear programming (NLP). A hybrid optimization algorithm based on PSO and SQP is proposed to solve the new NLP problem. PSO is used to search the initial value efficiently and guarantee the global performance of the initial value and SQP is used for local optimization based on initial value, which can give full play to the excellent global optimization performance of PSO and the outstanding local optimization performance of SQP. The effectiveness of the proposed optimization strategy is verified by a simulation example of the main belt asteroid Metis (1974 QU2) rendezvous mission with once Mars gravity assist. The results show that the optimal control curve obtained is in accordance with the optimal control form of bang-bang control derived from the theory, and the thruster has just two situations at any time: shutdown or startup with maximum thrust. In the working power range, increasing the number of thrust stages can improve the rate of energy utilization, thereby improving the efficiency of thruster, shortening the startup time and reducing fuel consumption. Meanwhile, ineffective grading should be avoided in the non-working power range.

Optimization of Satellite Orbits with Resonant Cross-Track Control

This paper is dedicated to developing a new approach for solving the low-thrust trajectory optimization problem of spacecraft in low Earth orbits. The unknown control function is parameterized by its Fourier coefficients up to a given order, thus transforming the optimal control problem into a nonlinear parameter optimization problem. An adequate choice of the period and the maximum degree of the truncated Fourier expansion leads to the creation of a resonance between the natural frequency of the orbital dynamics and the control function. Resonance is a well-known phenomenon; however, the novelty of the presented approach is to create the resonance artificially, in order to increase the effect of the control acceleration on the satellite trajectory evolution. This artificial resonance contributes to creating a rapid change of the cross-track orbital elements, using the low-thrust control acceleration. The proposed technique is applied to low Earth orbits constellation optimization, involving the minimization of the geometric dilution of precision of an Earth coverage problem.

Dynamical structures in a low-thrust, multi-body model with applications to trajectory design

A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously detail the evolution of the spacecraft position and velocity vectors, as well as the thrust history. To address this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP-LT) are explored as candidate solutions to seed initial low-thrust trajectory designs. Furthermore, insights from dynamical systems theory are leveraged to inform the design process. In the combined model, the addition of a low-thrust force modifies the locations and stability of the equilibria, resulting in flow configurations that differ from the natural behavior in the CR3BP. Families of periodic solutions in the vicinity of the equilibria supply novel geometries that may be employed in initial designs. Additionally, the application of simplifying assumptions yields a conservative, autonomous system with properties that supply useful insights. “Forbidden regions” at fixed energy levels bound low-thrust motion in such a simplified system, and analytical equations are available to guide the navigation through energy space. Periodic orbits and their associated manifolds also possess useful properties and act similarly to separatrices in the simplified regime. These structures and insights are employed to design transit and capture trajectories in the Earth-Moon CR3BP-LT.

Hybrid Differential Dynamic Programming in the Circular Restricted Three-Body Problem

This paper presents the application of differential dynamic programming to low-thrust trajectory optimization in the three-body problem along with supporting techniques. A penalty method is suggested for path inequality constraints and used to enforce a minimum radius constraint. Time variable formulations are presented for the flight time and a moving target. The latter is used to access all points along a desired periodic orbit as valid terminal states. Multiphase differential dynamic programming is applied to discontinuous initial guesses that combine segments of different periodic orbits. Numerical examples include multirevolution transfers between distant retrograde orbits, a heteroclinic transfer between planar Lyapunov orbits, and transfers between L1 and L2 halo orbits in the Earth–moon circular restricted three-body problem.

Mission design for multi-target and multi-mode rendezvous missions to small bodies

Multi-target exploration to small bodies has the advantages of reducing mission cost and increasing scientific return. It will be widely used in the future small body exploration missions. This paper investigates the problem of perturbation sensitivity, complicated constraints and changeable thrust, which may be faced in the multi-target exploration mission. The Earth quasi-satellite sampling return and main-belt comet rendezvous missions are studied. First, based on the mission constraints, the mission design schedule and relevant constraints are given. Then, the multi-impulse asteroid sampling return trajectory is investigated in the elliptical three body model. Considering the gravitational perturbation of Earth, the integral trajectory design consist of Earth escape phase, interplanetary transfer phase, and Earth reentry phase is proposed. The trajectory design is divided into two parts. The optimal preliminary transfer opportunity is firstly searched in two-body problem and is transferred to the elliptical three body model. The differential correction scheme is applied to correct the error caused by Earth gravity perturbation. The trajectory for the Earth quasi-satellite 2016HO3 is designed. Finally, considering the engineering constraints, the first-order necessary condition of the changeable low-thrust trajectory are derived, where both the thrust and special impulse change with the distance to Sun. By introducing the B-plane parameters, the initial states determination under the constraint of the gravity assist is developed. Based on the homotopy method, the low-thrust optimal trajectory after Mars gravity assist to the main belt comet 133P are designed and compared with constant low thrust trajectory. The proposed multi-target exploration scheme and the orbit design method can provide a reference for future small bodies exploration missions in China.

Linearization Methods for Problems of Optimizing Low-Thrust Spacecraft Trajectories: Test Problem Solutions

Several test problems are solved with the modified linearization method (MLM) demonstrating its features and possibilities.