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 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.展开更多
The low-thrust trajectory optimization with complicated constraints must be considered in practical engineering. In most literature, this problem is simplified into a two-body model in which the spacecraft is subject ...The low-thrust trajectory optimization with complicated constraints must be considered in practical engineering. In most literature, this problem is simplified into a two-body model in which the spacecraft is subject to the gravitational force at the center of mass and the spacecraft＇s own electric propulsion only, and the gravity assist （GA） is modeled as an instantaneous velocity increment. This paper presents a method to solve the fuel-optimal problem of low-thrust trajectory with complicated constraints in a full ephemeris model, which is closer to practical engineering conditions. First, it introduces various perturbations, including a third body＇s gravity, the nonspherical perturbation and the solar radiation pressure in a dynamic equation. Second, it builds two types of equivalent inner constraints to describe the GA. At the same time, the present paper applies a series of techniques, such as a homotopic approach, to enhance the possibility of convergence of the global optimal solution.展开更多
Spacecraft science missions to planets or asteroids have historically visited only one or several celestial bodies per mission.The research goal of this paper is to create a trajectory design algorithm that generates ...Spacecraft science missions to planets or asteroids have historically visited only one or several celestial bodies per mission.The research goal of this paper is to create a trajectory design algorithm that generates trajectory allowing a spacecraft to visit a significant number of asteroids during a single mission.For the problem of global trajectory optimization,even with recent advances in low-thrust trajectory optimization,a full enumeration of this problem is not possible.This work presents an algorithm to traverse the searching space in a practical fashion and generate solutions.The flight sequence is determined in ballistic scenario,and a differential evolution method is used with constructing a three-impulse transfer problem,then the local optimization is implemented with low-thrust propulsion on the basis of the solutions of impulsive trajectories.The proposed method enables trajectory design for multiple asteroids tour by using available low thrust propulsion technology within fuel and time duration constraints.展开更多
A modified direct optimization method is proposed to solve the optimal multi-revolution transfer with low-thrust between Earth-orbits. First, through parameterizing the control steering angles by costate variables, th...A modified direct optimization method is proposed to solve the optimal multi-revolution transfer with low-thrust between Earth-orbits. First, through parameterizing the control steering angles by costate variables, the search space of free parameters has been decreased. Then, in order to obtain the global optimal solution effectively and robustly, the simulated annealing and penalty function strategies were used to handle the constraints, and a GA/SQP hybrid optimization algorithm was utilized to solve the parameter optimization problem, in which, a feasible suboptimal solution obtained by GA was submitted as an initial parameter set to SQP for refinement. Comparing to the classical direct method, this novel method has fewer free parameters, needs not initial guesses, and has higher computation precision. An optimal-fuel transfer problem from LEO to GEO was taken as an example to validate the proposed approach. The results of simulation indicate that our approach is available to solve the problem of optimal muhi-revolution transfer between Earth-orbits.展开更多
This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium point...This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium points (AEPs) are generated by canceling the gravitational and centrifugal forces with continuous low-thrust at a non-equilibrium point. Some graphical investigations are shown for the effects of the relative parameters which characterized the locations of the AEPs. Also, the numerical values of AEPs have been calculated. The positions of these AEPs will depend not only also on magnitude and directions of low-thrust acceleration. The linear stability of the AEPs has been investigated. We have determined the stability regions in the xy, xz and yz-planes and studied the effect of oblateness parameters A1(0A1?and ?A2(0A2<1) on the motion of the spacecraft. We have found that the stability regions reduce around both the primaries for the increasing values of oblateness of the primaries. Finally, we have plotted the zero velocity curves to determine the possible regions of motion of the spacecraft.展开更多
Low-thrust Earth-orbit transfers with 10^- 5-order thrust-to-weight ratios involve a large number of orbital revolutions which poses a real challenge to trajectory optimization. This article develops a direct method t...Low-thrust Earth-orbit transfers with 10^- 5-order thrust-to-weight ratios involve a large number of orbital revolutions which poses a real challenge to trajectory optimization. This article develops a direct method to optimize minimum-time low-thrust many-revolution Earth-orbit transfers. A parameterized control law in each orbit, in the form of the true optimal control, is proposed, and the time history of the parameters governing the control law is interpolated through a finite number of nodal values. The orbital averaging method is used to significantly reduce the computational workload and the trajectory optimization is conducted based on the orbital averaging dynamics expressed by nonsingular equinoctial elements. Furthermore, Earth＇s shadowing and perturbation effects are taken into account. The optimal transfer problem is thus converted to the parameter optimization problem that can be solved by nonlinear programming. Taking advantage of the mapping between the parameterized control law and the Lyapunov control law, a technique is proposed to acquire good initial guesses for optimization variables, which results in enlarged convergence domain of the direct optimization method. Numerical examples of optimal Earth-orbit transfers are presented.展开更多
Future missions to the Moon and beyond are likely to involve low-thrust propulsion technologies due to their propellant efficiency.However,these still present a difficult trajectory design problem,owing to the near co...Future missions to the Moon and beyond are likely to involve low-thrust propulsion technologies due to their propellant efficiency.However,these still present a difficult trajectory design problem,owing to the near continuous thrust,lack of control authority and chaotic dynamics.Lyapunov control laws can generate sub-optimal trajectories for such missions with minimal computational cost and are suitable for feasibility studies and as initial guesses for optimisation methods.In this work a Reinforced Lyapunov Controller is used to design optimal low-thrust transfers from geostationary transfer orbit towards lunar polar orbit.Within the reinforcement learning(RL)framework,a dual-actor network setup is used,one in each of the Earth-and Moon-centred inertial frames respectively.A key contribution of this paper is the demonstration of a forwards propagated trajectory,removing the need to define a patch point a priori.This is enabled by an adaptive patch distance and extensive initial geometry exploration during the RL training.Results for both time-and fuel-optimal transfers are presented,along with a Monte Carlo analysis of the robustness to disturbances for such transfers.Phasing is introduced where necessary to aid rendezvous with the Moon.The results demonstrate the potential for such techniques to provide a basis for the design and guidance of low-thrust lunar transfers.展开更多
Optimization of low-thrust trajectories that involve a larger number of orbit revolutions is considered as a challenging problem.This paper describes a high-precision symplectic method and optimization techniques to s...Optimization of low-thrust trajectories that involve a larger number of orbit revolutions is considered as 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.展开更多
This paper develops a sequential convex programming(SCP)-based method to solve the minimum-fuel variable-specific-impulse low-thrust transfer problem considering shutdown constraint,with emphasize on improving the com...This paper develops a sequential convex programming(SCP)-based method to solve the minimum-fuel variable-specific-impulse low-thrust transfer problem considering shutdown constraint,with emphasize on improving the computational efficiency.The variable parameter engine is more applicable for many low-thrust scenarios,therefore,both a continuously variable model and a ladder variable model are adopted.First,the original problem is convexified by processing the constraint feasible domain,which is composed of the nonlinear dynamic equations and second-order equality constraint,into convex sets.Then,the approximation is generated to close the optimal solution of the low-thrust problem by iteratively solving the convexified subproblem.Moreover,the switching self-detection and adaptive node refinement methods are presented,which can improve the accuracy of the solution and accelerate the convergence during the approximation process and is especially necessary and effective in the scenarios with shutdown constraint.In numerical simulations,the comparison with the homotopic approach shows that the proposed method only needs 4%computational time as that of the homotopic approach,and two variable-specificimpulse examples further demonstrate the effectiveness and efficiency of the proposed method.展开更多
The preliminary mission design of spacecraft missions to asteroids often involves,in the early phases,the selection of candidate target asteroids.The final result of such an analysis is a list of asteroids,ranked with...The preliminary mission design of spacecraft missions to asteroids often involves,in the early phases,the selection of candidate target asteroids.The final result of such an analysis is a list of asteroids,ranked with respect to the necessary propellant to be used,that the spacecraft could potentially reach.In this paper we investigate the sensitivity of the produced asteroids rank to the employed trajectory model in the specific case of a small low-thrust propelled spacecraft beginning its journey from the Sun–Earth L2 Lagrangian point and heading to a rendezvous with some near-Earth asteroid.We consider five increasingly complex trajectory models:impulsive,Lambert,nuclear electric propulsion,nuclear electric propulsion including the Earth’s gravity,solar electric propulsion including the Earth’s gravity and we study the final correlation between the obtained target rankings.We find that the use of a lowthrust trajectory model is of great importance for target selection,since the use of chemical propulsion surrogates leads to favouring less attractive options 19%of times,a percentage that drops to 8%already using a simple nuclear electric propulsion model that neglects the Earth’s gravity effects and thrust dependence on the solar distance.We also find that for the study case considered,a small interplanetary CubeSat named M-ARGO,the inclusion of the Earth’s gravity in the considered dynamics does not affect the target selection significantly.展开更多
基金supported by the National Natural Science Foundation of China(Nos.61627810,61790562,61403096).
文摘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.
文摘The low-thrust trajectory optimization with complicated constraints must be considered in practical engineering. In most literature, this problem is simplified into a two-body model in which the spacecraft is subject to the gravitational force at the center of mass and the spacecraft＇s own electric propulsion only, and the gravity assist （GA） is modeled as an instantaneous velocity increment. This paper presents a method to solve the fuel-optimal problem of low-thrust trajectory with complicated constraints in a full ephemeris model, which is closer to practical engineering conditions. First, it introduces various perturbations, including a third body＇s gravity, the nonspherical perturbation and the solar radiation pressure in a dynamic equation. Second, it builds two types of equivalent inner constraints to describe the GA. At the same time, the present paper applies a series of techniques, such as a homotopic approach, to enhance the possibility of convergence of the global optimal solution.
文摘Spacecraft science missions to planets or asteroids have historically visited only one or several celestial bodies per mission.The research goal of this paper is to create a trajectory design algorithm that generates trajectory allowing a spacecraft to visit a significant number of asteroids during a single mission.For the problem of global trajectory optimization,even with recent advances in low-thrust trajectory optimization,a full enumeration of this problem is not possible.This work presents an algorithm to traverse the searching space in a practical fashion and generate solutions.The flight sequence is determined in ballistic scenario,and a differential evolution method is used with constructing a three-impulse transfer problem,then the local optimization is implemented with low-thrust propulsion on the basis of the solutions of impulsive trajectories.The proposed method enables trajectory design for multiple asteroids tour by using available low thrust propulsion technology within fuel and time duration constraints.
基金Sponsored by the National Natural Science Foundation of China(Grant No.10672044)
文摘A modified direct optimization method is proposed to solve the optimal multi-revolution transfer with low-thrust between Earth-orbits. First, through parameterizing the control steering angles by costate variables, the search space of free parameters has been decreased. Then, in order to obtain the global optimal solution effectively and robustly, the simulated annealing and penalty function strategies were used to handle the constraints, and a GA/SQP hybrid optimization algorithm was utilized to solve the parameter optimization problem, in which, a feasible suboptimal solution obtained by GA was submitted as an initial parameter set to SQP for refinement. Comparing to the classical direct method, this novel method has fewer free parameters, needs not initial guesses, and has higher computation precision. An optimal-fuel transfer problem from LEO to GEO was taken as an example to validate the proposed approach. The results of simulation indicate that our approach is available to solve the problem of optimal muhi-revolution transfer between Earth-orbits.
文摘This paper studies the existence and stability of the artificial equilibrium points (AEPs) in the low-thrust restricted three-body problem when both the primaries are oblate spheroids. The artificial equilibrium points (AEPs) are generated by canceling the gravitational and centrifugal forces with continuous low-thrust at a non-equilibrium point. Some graphical investigations are shown for the effects of the relative parameters which characterized the locations of the AEPs. Also, the numerical values of AEPs have been calculated. The positions of these AEPs will depend not only also on magnitude and directions of low-thrust acceleration. The linear stability of the AEPs has been investigated. We have determined the stability regions in the xy, xz and yz-planes and studied the effect of oblateness parameters A1(0A1?and ?A2(0A2<1) on the motion of the spacecraft. We have found that the stability regions reduce around both the primaries for the increasing values of oblateness of the primaries. Finally, we have plotted the zero velocity curves to determine the possible regions of motion of the spacecraft.
基金National Natural Science Foundation of China (10603005)
文摘Low-thrust Earth-orbit transfers with 10^- 5-order thrust-to-weight ratios involve a large number of orbital revolutions which poses a real challenge to trajectory optimization. This article develops a direct method to optimize minimum-time low-thrust many-revolution Earth-orbit transfers. A parameterized control law in each orbit, in the form of the true optimal control, is proposed, and the time history of the parameters governing the control law is interpolated through a finite number of nodal values. The orbital averaging method is used to significantly reduce the computational workload and the trajectory optimization is conducted based on the orbital averaging dynamics expressed by nonsingular equinoctial elements. Furthermore, Earth＇s shadowing and perturbation effects are taken into account. The optimal transfer problem is thus converted to the parameter optimization problem that can be solved by nonlinear programming. Taking advantage of the mapping between the parameterized control law and the Lyapunov control law, a technique is proposed to acquire good initial guesses for optimization variables, which results in enlarged convergence domain of the direct optimization method. Numerical examples of optimal Earth-orbit transfers are presented.
文摘Future missions to the Moon and beyond are likely to involve low-thrust propulsion technologies due to their propellant efficiency.However,these still present a difficult trajectory design problem,owing to the near continuous thrust,lack of control authority and chaotic dynamics.Lyapunov control laws can generate sub-optimal trajectories for such missions with minimal computational cost and are suitable for feasibility studies and as initial guesses for optimisation methods.In this work a Reinforced Lyapunov Controller is used to design optimal low-thrust transfers from geostationary transfer orbit towards lunar polar orbit.Within the reinforcement learning(RL)framework,a dual-actor network setup is used,one in each of the Earth-and Moon-centred inertial frames respectively.A key contribution of this paper is the demonstration of a forwards propagated trajectory,removing the need to define a patch point a priori.This is enabled by an adaptive patch distance and extensive initial geometry exploration during the RL training.Results for both time-and fuel-optimal transfers are presented,along with a Monte Carlo analysis of the robustness to disturbances for such transfers.Phasing is introduced where necessary to aid rendezvous with the Moon.The results demonstrate the potential for such techniques to provide a basis for the design and guidance of low-thrust lunar transfers.
基金supported by the National Natural Science Foundation of China(Grant Nos.11672146,11432001)the 2015 Chinese National Postdoctoral International Exchange Program
文摘Optimization of low-thrust trajectories that involve a larger number of orbit revolutions is considered as 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.
基金supported by the National Key R&D Program of China(Grant No.2020YFC2201200)the National Natural Science Foundation of China(Grant No.U20B2001)。
文摘This paper develops a sequential convex programming(SCP)-based method to solve the minimum-fuel variable-specific-impulse low-thrust transfer problem considering shutdown constraint,with emphasize on improving the computational efficiency.The variable parameter engine is more applicable for many low-thrust scenarios,therefore,both a continuously variable model and a ladder variable model are adopted.First,the original problem is convexified by processing the constraint feasible domain,which is composed of the nonlinear dynamic equations and second-order equality constraint,into convex sets.Then,the approximation is generated to close the optimal solution of the low-thrust problem by iteratively solving the convexified subproblem.Moreover,the switching self-detection and adaptive node refinement methods are presented,which can improve the accuracy of the solution and accelerate the convergence during the approximation process and is especially necessary and effective in the scenarios with shutdown constraint.In numerical simulations,the comparison with the homotopic approach shows that the proposed method only needs 4%computational time as that of the homotopic approach,and two variable-specificimpulse examples further demonstrate the effectiveness and efficiency of the proposed method.
文摘The preliminary mission design of spacecraft missions to asteroids often involves,in the early phases,the selection of candidate target asteroids.The final result of such an analysis is a list of asteroids,ranked with respect to the necessary propellant to be used,that the spacecraft could potentially reach.In this paper we investigate the sensitivity of the produced asteroids rank to the employed trajectory model in the specific case of a small low-thrust propelled spacecraft beginning its journey from the Sun–Earth L2 Lagrangian point and heading to a rendezvous with some near-Earth asteroid.We consider five increasingly complex trajectory models:impulsive,Lambert,nuclear electric propulsion,nuclear electric propulsion including the Earth’s gravity,solar electric propulsion including the Earth’s gravity and we study the final correlation between the obtained target rankings.We find that the use of a lowthrust trajectory model is of great importance for target selection,since the use of chemical propulsion surrogates leads to favouring less attractive options 19%of times,a percentage that drops to 8%already using a simple nuclear electric propulsion model that neglects the Earth’s gravity effects and thrust dependence on the solar distance.We also find that for the study case considered,a small interplanetary CubeSat named M-ARGO,the inclusion of the Earth’s gravity in the considered dynamics does not affect the target selection significantly.
