Low-thrust Trajectory Optimization Research Articles

Robust Low-Thrust Trajectory Optimization Using Convex Programming and a Homotopic Approach

A robust algorithm to solve the low-thrust fuel-optimal trajectory optimization problem for interplanetary spacecraft is developed in this article. The original nonlinear optimal control problem is convexified and transformed into a parameter optimization problem using an arbitrary-order Gauss–Lobatto discretization scheme with nonlinear control interpolation. A homotopic approach that considers the energy-to-fuel smoothing path is combined with an adaptive second-order trust-region mechanism to increase performance. The overall robustness is assessed in several fuel-optimal transfers with poor initial guesses. The results show a superior performance in terms of convergence and computational time compared to standard convex programming approaches in the literature.

Optimal Escape from Sun-Earth and Earth-Moon L2 with Electric Propulsion

Optimal low-thrust trajectories for the direct escape from the Earth’s sphere of influence, starting from Sun-Earth or Earth-Moon L2, are analyzed with an indirect optimization method. The dynamic model considers four-body gravitation and JPL ephemeris; solar radiation pressure is also considered. Specific techniques and improvements to the method are introduced to tackle the highly chaotic and nonlinear dynamics of motion close to Lagrangian points, which challenges the remarkable precision of the indirect method. The results show that escape trajectories have optimal performance when the solar perturbation acts favorably in both thrust and coast phases. The effects of the solar and Moon perturbations are more evident in the Earth-Moon L2 escapes compared with those from the Sun-Earth L2. EML2 escapes have single- or two-burn solutions depending on the trajectory deflection, which is needed to have a favorable solar perturbation. The SEL2 escapes, on the contrary, mainly have a single initial burn and a long coast arc, but need an additional final thrust arc if the required C3 is high. As applications of such Lagrangian Point trajectories, results include considerations about escape maneuvers from different SEL2 high-fidelity Lyapunov orbits and escape for interplanetary trajectories towards near-earth asteroids.

Optimization of low-thrust multi-debris removal mission via an efficient approximation model of orbit rendezvous

An efficient iterative approximation model for low-thrust rendezvous in low-earth orbit is proposed to be applied in optimization of multi-debris removal missions. Based on an existing method for impulsive orbit rendezvous and a novel iteration process, the propellant cost and thrust time of a low-thrust rendezvous can be quickly evaluated. Moreover, another iterative method is also designed to help find the minimal transfer duration required for orbit rendezvous. Then, its application in the max-reward optimization of debris removal mission with a large-scale targets to be selected is studied. In the three-steps optimization including target selection, sequence optimization, and low-thrust trajectory optimization, the iterative approximation model is used to deal with the constraints and evaluate the objective functions more efficiently. The problem of the eighth Chinese trajectory optimization competition was used to test the performance, and a leading solution including 93 targets was found. Comparisons with previous results were made to prove the advantages of the method in this paper.

Multi-Objective Low-Thrust Trajectory Optimization with Robustness to Missed Thrust Events

This paper introduces a new technique for directly controlling the missed thrust recovery margin (MTRM) of a low-thrust spacecraft trajectory. MTRM is defined here as the longest amount of time a spacecraft may coast away from a nominal trajectory while still being able to reach a terminal manifold once thruster operations are resumed. The “virtual swarm” optimization technique developed here simultaneously optimizes the nominal spacecraft trajectory along with many recovery trajectories. The objective can be to maximize the MTRM of the nominal trajectory at its weakest point or to constrain the worst-case MTRM to be at or above a desired level while optimizing a different value (e.g., mass, time of flight). The technique is demonstrated for a direct Earth–Mars transfer and for a gravity assist trajectory to the asteroid Psyche. Further, a method for finding the Pareto front of MTRM, arrival mass, and arrival date is presented to address the related multi-objective optimization problem.

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.

Analytical Initialization for Low-thrust Trajectory Optimization Based on Switching System

The traditional homotopy method usually transforms the low-thrust fuel-optimal control problem into the energy-optimal problem to increase the convergence rate of the indirect method. However, it is still necessary to guess the initial values of the co-states to initialize the solving algorithm. In this paper, the optimization model of the fuel-optimal problem is embedded in the switching system with the analytical initial co-states, which further improves the convergence rate with the analytical initialization. Firstly, the switching system is introduced with the embedded fuel-optimal problem. The switching function of the conventional switching system is derived from the optimal control, but in this paper, the given switching function is designed artificially to realize the switching and continuation among different systems. Secondly, based on the linearization technique, the target system is designed with analytical initial co-states, initializing the solving algorithm by a simple nominal trajectory. Finally, the numerical simulation verifies the effectiveness of the proposed method, which is more efficient than the traditional homotopy method.

Combined System–Trajectory Design for Geostationary Orbit Platforms on Hybrid Transfer

A novel methodology for a combined systems–trajectory optimization for a geostationary equatorial orbit (GEO) platform is proposed to obtain comprehensive design solutions. A combined chemical–electric propulsion system is used to execute hybrid high-thrust/low-thrust trajectory transfer to GEO, thereby balancing the overall system mass and transfer time. A systematic and payload-centric mission design provides a new set of design options to deliver tailored solutions to customized payloads. The hybrid trajectory characterization and spacecraft systems design find the required platform launch mass to deliver a GEO platform with a defined final mass and operational power. Elements of the system design are combined with those of multispiral low-thrust trajectory optimization as well as radiation absorption and solar array degradation to provide a comprehensive design solution. The result is a wide set of solutions to reach GEO, where fully chemical and fully electric transfers represent the boundaries of the hybrid transfer trade space. A payload throughput power of 20 kW entails a spacecraft mass in GEO between 4000 and 4550 kg, an initial thrust-to-mass ratio range of , and a cover-glass thickness between 4 and 24 mils to guarantee a minimum end-of-life/beginning-of-life power ratio of 85%. In addition, all-electric solutions from different injection orbits yield transfers to GEO with a time of flight of 60–150 days and an initial mass for the platform of 4400–5500 kg.

Indirect low-thrust trajectory optimization with gridded ion thruster model

The work deals with indirect optimization of minimum-time and minimum-fuel interplanetary trajectories when gridded ion thruster models are considered. Using an accurate model of solar electric propulsion is beneficial in preliminary mission design, and allows including operational constraints. The maximum thrust and the specific impulse are expressed as a function of thruster input power, which is achieved by means of point-fitting lines that match the performance capabilities of the thrusters. Minimum-time and minimum-fuel problems are formulated to be solved by indirect optimization. In order to increase the accuracy and robustness of the shooting procedure, analytic Jacobians are derived, and a hybrid switching detection technique is used to improve the integration accuracy for minimum-fuel problems. Two examples of Earth-to-Mars transfer and Near-Earth rendezvous mission using the realistic NASA’s Evolutionary Xenon Thruster (NEXT) are given to substantiate the feasibility and effectiveness of the proposed method.

Stochastic Primer Vector for Robust Low-Thrust Trajectory Design Under Uncertainty

Any space trajectories are subject to state uncertainty due to imperfect state knowledge, random disturbances, and partially known dynamical environments. Ideally, such uncertainty and associated risks must be properly quantified and taken into account in the process of trajectory design, ensuring a sufficiently low risk of causing hazardous events. To bridge the gap between the ideal goal and current practice in mission design, this paper extends Lawden’s primer vector theory and develops a solution method to solve the problem of low-thrust trajectory optimization under state uncertainty. The new primer vector, termed stochastic primer vector, provides an analytical open-loop optimal control law that respects a probabilistic path constraint with a user-specified confidence level (chance constraint). A numerical aspect of the indirect method is also extended by introducing a smoothing approach across constraint corner discontinuities, enabling efficient solution methods for constrained optimal control problems. The validity and effectiveness of the theoretical development are demonstrated with two numerical examples, which clarify the behavior of chance-constrained optimal low-thrust trajectories and confirm through Monte Carlo simulations that the designed trajectories indeed satisfy the imposed constraints under uncertainty with the prescribed confidence level.

Optimization of Multi-revolution Limited Power Trajectories Using Angular Independent Variable

Optimization of low-thrust trajectories is necessary in the design of space missions using electric propulsion systems. We consider the problem of limited power trajectory optimization, which is a well-known case of the low-thrust optimization problem. In the article, we present an indirect approach to trajectory optimization based on the use of the maximum principle and the continuation method. We introduce the concept of auxiliary longitude and use it as a new independent variable instead of time. The use of equations of motion in the equinoctial elements and a new independent variable allowed us to simplify the optimization of limited power trajectories with a fixed angular distance and free transfer duration. The article presents a new form of necessary optimality conditions for this problem and describes an efficient new numerical method to solve the limited power trajectory optimization problem. We show the existence of several trajectories with a fixed transfer duration and free angular distance that satisfy the necessary optimality conditions. Using numerical examples, we confirm the existence of the limiting values of the characteristic velocity and the product of the cost function value and the transfer duration as the angular distance increases. The high computational performance of the developed technique makes it possible to carry out and present an analysis of the angular flight range and initial true longitude impact on the cost function, transfer duration, and characteristic velocity.

A Survey on Low-Thrust Trajectory Optimization Approaches

In this paper, we provide a survey on available numerical approaches for solving low-thrust trajectory optimization problems. First, a general mathematical framework based on hybrid optimal control will be presented. This formulation and their elements, namely objective function, continuous and discrete state and controls, and discrete and continuous dynamics, will serve as a basis for discussion throughout the whole manuscript. Thereafter, solution approaches for classical continuous optimal control problems will be briefly introduced and their application to low-thrust trajectory optimization will be discussed. A special emphasis will be placed on the extension of the classical techniques to solve hybrid optimal control problems. Finally, an extensive review of traditional and state-of-the art methodologies and tools will be presented. They will be categorized regarding their solution approach, the objective function, the state variables, the dynamical model, and their application to planetocentric or interplanetary transfers.

High-precision shape approximation low-thrust trajectory optimization method satisfying bi-objective index

The shape approximation method has been proven to be rapid and practicable in resolving low-thrust trajectory; however, it still faces the challenges of large deviation from the optimal solution and inability to satisfy the specific flight time and fuel mass constraints. In this paper, a modified shape approximation low-thrust model is presented, and a novel constrained optimization algorithm is developed to solve this problem. The proposed method aims at settling the bi-objective optimization orbit involving the twin objectives of minimum flight time and low fuel consumption and enhancing the accuracy of optimized orbit. In particular, a transformed high-order polynomial model based on finite Fourier series is proposed, which can be characterized as a multi-constraint optimization problem. Then, a novel optimization algorithm is specifically developed to optimize the large-scale multi-constraint dynamical equations of shape trajectory. The key performance indicators of the index include minimum flight time, low fuel consumption and bi-objective optimization of the two. Simulation results prove that this approach possesses both the high precision achievable by numerical methods and low computational complexity offered by shape approximation techniques. Besides, the Pareto front of the fuel-time bi-objective optimization orbit is firstly introduced to analyze an intact optimal solution set. Furthermore, we have demonstrated that our proposed approach is appropriate to generate the preliminary orbit for pseudo-spectral method.

Low-thrust transfer to the Earth-Moon triangular libration point via horseshoe orbit

This paper proposes a novel method for the low-thrust transfer to the triangular libration points in the Earth-Moon system. Horseshoe orbit is explored as the coasting arc in this transfer scenario. By tangential thrust control strategy, the low-thrust transfer from the Earth to the short-period orbit around L4 point is realized. The pseudospectral method is used to obtain another Earth-L4 transfer trajectory by solving an energy-optimal low-thrust trajectory optimization problem. Interestingly, these two kinds of Earth-L4 low-thrust transfer trajectories both share similar geometric configuration and velocity trend with a natural Horseshoe orbit, which evokes the main contributions of this paper that leveraging Horseshoe orbit as the coasting arc. The complete transfer trajectory is divided into three phases: Earth spiral phase, Horseshoe coasting phase and L4 point insertion phase, where the first and the third are powered phases. The tangential thrust control strategy is used in the first phase to generate the transfer from the Earth to the Horseshoe orbit, and the pseudospectral method is utilized in the third phase to enable the transfer from the Horseshoe coasting arc to the L4 point by solving a time-optimal transfer problem. Then, by patching these two powered segments with the unpowered Horseshoe coasting arc, the Earth-L4 low-thrust transfer is constructed.

Low-thrust spacecraft trajectory optimization via a DNN-based method

Initial solution guess has a significant impact on the convergence of indirect methods, especially for continuous low-thrust trajectory optimization problems. In this study, an intelligent initial solution supplying approach based on deep neural networks (DNNs) is proposed to help achieve the fast generation of optimal trajectories for low-thrust orbit transfers. Energy-optimal and fuel-optimal trajectories with three different terminal conditions are considered. Based on the training dataset obtained by an indirect method, DNNs are constructed to approximate the solutions corresponding to different flight states. Based on the trained DNNs, an intelligent trajectory optimization method named DNN-based method is developed with the help of the homotopy technique. Numerical simulations are conducted to evaluate the performance of the proposed method on success rates and time consumptions. Simulation results demonstrate that the combination of traditional techniques and the new DNN technology can achieve the fast generation of low-thrust optimal trajectories with advantages on computational efficiency and reliability.

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.

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.

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.

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.

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.