Low-thrust Spacecraft Research Articles

Fast Trajectory Generation and Asteroid Sequence Selection in Multispacecraft for Multiasteroid Exploration.

As an increasing number of asteroids are being discovered, detecting them using limited propulsion resources and time has become an urgent problem in the aerospace field. However, there is no universal fast asteroid sequence selection method that finds the trajectories for multiple low-thrust spacecraft for detecting a large number of asteroids. Furthermore, the calculation efficiency of the traditional trajectory optimization method is low, and it requires a large number of iterations. Therefore, this study combines Monte Carlo tree search (MCTS) with spacecraft trajectory optimization. A fast MCTS pruning algorithm is proposed, which can quickly complete asteroid sequence selection and trajectory generation for multispacecraft exploration of multiple asteroids. By combining the Bezier shape-based (SB) method and MCTS, this study realizes the fast search of the exploration sequence and the efficient optimization of the continuous transfer trajectories. In the simulation example, compared with the traversal algorithm, the MCTS pruning algorithm obtained the global optimal detection sequence of the search tree in a very short time. Under the same conditions, the Bezier SB method obtained the transfer trajectory with a better performance index faster than the finite Fourier series SB method. Performances of the proposed method are illustrated through a complex asteroid multiflyby mission design.

Hybrid optimization of low-thrust many-revolution trajectories with coasting arcs and longitude targeting for propellant minimization

Despite the ongoing advancements in low-thrust propulsion technology and the rise of all-electric satellite platforms, low-thrust spacecraft trajectory optimization remains a complex field of research. Shape-based approximations are predominant in interplanetary applications, but they are generally unsuitable for many-revolution trajectories, common in terrestrial applications. Indirect optimization methods allow for global optimization of many-revolution trajectories, but their mathematical complexity generally requires significant simplifications of the dynamical model, and they must be re-derived for any modification to the system dynamics or constraints. Conversely, direct optimization methods exhibit larger convergence radii and are flexible for application in different problems yet suffer from impractical computational times due to large design vectors.This paper presents a methodology for the optimization of low-thrust many-revolution trajectories, employing a hybrid combination of indirect and direct optimization methods. Similar hybrid approaches have been shown to be highly reliable for minimum-time trajectories. This methodology preserves similar performance while additionally enabling minimum-propellant optimization, through a mechanism that allows for coasting (non-thrusting) arcs, as well as targeting of the final geodetic-longitude. To reduce the propagation load of the methodology, we combine an orbital averaging scheme with a differential evolution algorithm, leading to a global optimization process with a practical computational effort. The analytical nature of the methodology reduces the number of optimization variables and its computational counterpart provides unmatchable flexibility for a configurable force and perturbation model as well as operational constraints fulfilment.The approach is applied to an unperturbed and a J2-perturbed GTO-GEO transfer, revealing a 0.03% and a 0.4% error, for time- and propellant-minimization respectively, relative to the reference optimal trajectories. This proves that the method can match the performance of former hybrid approaches while additionally allowing for engine on/off switching. Moreover, the inclusion of the J2 perturbation shows that, in contrast to indirect methods, it can accommodate modifications to the system dynamics without the need to re-derive the optimal control laws. Furthermore, a superior convergence radius of the optimization problem is demonstrated for the hybrid method, with respect to a reference indirect method, through the simultaneous optimization for minimum-propellant expenditure and final geodetic-longitude targeting. This research constitutes a significant advancement for space mission design and satellite operations, because it simultaneously harnesses the advantages of indirect and direct methods with broader flexibility than the popular indirect approaches and enhanced functionality than the former hybrid methods published in literature.

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.

Warm-Start Multihomotopic Optimization for Low-Thrust Many-Revolution Trajectories

Traditionally, fuel-optimal problems for low-thrust spacecraft transfers are connected with energy-optimal problems to increase the convergence possibility of indirect methods. However, for low-thrust many-revolution missions, it is still hard to resolve the energy-optimal problems in a fast and reliable manner due to the strong nonlinearity. To address this issue, a warm-start multihomotopic optimization approach is proposed in this article, wherein the Sundman transformation and multihomotopic techniques are used to connect the energy-optimal problems to linear ones with analytical solutions. This article focuses on the following three contributions. First, the energy-optimal control problems are transformed into true longitude-depended energy-optimal problems based on the homotopy to dynamical model and Sundman transformation, and the equivalence relations of the solutions before and after transformation are obtained analytically. Second, the problems are further connected with time-free energy-optimal problems based on a homotopy to the time constraint. Third, the time-free problems are linearized near a nominal trajectory, and the corresponding analytical solutions are obtained. Starting with the analytical solutions, the algorithm can gradually iterate back to the solutions of the original energy- and fuel-optimal problems. Since the whole transformation process is lossless (optimality is preserved under the transformation process), the developed warm-start multihomotopic algorithm enjoys the advantages on reliable convergence and high computational efficiency. Numerical simulations of Earth-orbit transfer missions from GTO to GEO are conducted by comparing with traditional methods, and the results are given to substantiate the effectiveness of the proposed warm-start multihomotopic method.

Q-Law Aided Direct Trajectory Optimization of Many-Revolution Low-Thrust Transfers

Low-thrust spacecraft trajectory optimization for the many-revolution orbital transfer problem is especially challenging due to the high problem dimension and perturbing accelerations that prevent or complicate accurate analytical solutions. This analysis seeks to simplify the calculation of the nonaveraged spiral trajectory through the use of the well-known Q-Law guidance algorithm. Here, Q-Law is used to seed phases of the direct trajectory optimization problem. We explore the trade space between the fast calculation speed of Q-Law and the time or mass optimality of the full optimization problem using the common example of a geostationary transfer orbit to geostationary transfer. We find that a computationally efficient, near-optimal solution can be achieved by using Q-Law for an initial fraction of a spiral trajectory and direct optimization for the remainder of the trajectory. This reduces the direct optimization problem to a lower dimension and enables the spacecraft to reach a specific final state. Q-Law is particularly effective for portions of the trajectory where there are many eclipses, which can challenge direct optimization methods.

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.

Comparison of the Legendre–Gauss pseudospectral and Hermite–Legendre–Gauss–Lobatto methods for low-thrust spacecraft trajectory optimization

Low-thrust spacecraft propulsion systems enable fuel-efficient trajectories through space but the resulting trajectory optimization problems can be challenging. Such problems often use direct collocation methods for transcribing the optimal control problem into a nonlinear programming problem. In this work, the Hermite–Legendre–Gauss–Lobatto (HLGL) and the Legendre–Gauss pseudospectral (PS) direct collocation methods have been compared for a minimum-time low-thrust Earth-to-Mars transfer problem. It is demonstrated that trajectories with different time domains can be effectively compared using points that correspond to the same normalized times. Various metrics that describe the computational cost as well as the errors with respect to a reference trajectory have been used. For the HLGL method, the number of subintervals (m) and the order of the interpolating polynomial (n), can be varied such that the number of nodes (N) stays constant. The distribution of the number of such (m, n) pairs is analysed using the concept of divisor functions from number theory and an algorithm for determining all the possible combinations for any N value is developed. In addition, a method is presented by which the Edelbaum trajectory is used to set bounds, scale the problem, and generate a suitable initial guess. The primary contribution of this paper involves a detailed comparison of the performance of all the HLGL pairs with the PS method for nodes in the range $$N = 6$$ to $$N = 40$$ for the minimum-time low-thrust Earth-to-Mars transfer.

A New Heuristic Approach for Low-Thrust Spacecraft Trajectory Optimization

Electrical propulsion is gaining popularity in the satellite industry. Their high efficiency however comes at the cost of decreased thrust and longer transfer times. As such the interest in solving the large optimization problems associated with these long transfers has re-surged. This paper presents a heuristic approach to solving such low-thrust satellite trajectory optimization problems, based on control law blending and averaging. We derive the nonlinear program resulting from this approach and introduce PANOC, a recent optimization algorithm well suited for optimal control problems. The resulting controller is considerably faster than more classical, non-heuristic approaches, without a significant loss in optimality.

Investigation of Low Thrust Optimal Orbital Transfer from LEO to GEO Considering Circular Orbits

In the present research, optimal control problem for low thrust spacecraft orbital transfer is investigated by combination of analytical method and Artificial Bee Colony (ABC) algorithm, which is a population-based method. Spacecraft trajectory optimization is defined for orbital transfer from Low Earth Orbit (LEO) to Geosynchronous Earth Orbit (GEO). In the proposed analytical method co-state variables are simplified by employing concepts of optimal control and specific conditions which are considered for the problem, consequently, an adjoint differential equation is added to set of differential equations and these equations are solved simultaneously that result in optimum transfer trajectory. Dynamic modeling of the problem is represented based on Edelbaum equations and trajectory design is carried out considering circular orbits. Optimization problem is defined for different kinds of objective functions such as minimum time, minimum control effort, minimum time with minimum control effort as well as both minimum time and minimum control effort considering different optimal weighted coefficients. These coefficients are searched by means of artificial bee colony algorithm in weighted minimum time minimum effort scheme. Initial values of control variable and terminal time for all defined objective functions are foraged through artificial bee colony algorithm. The rate of changes of the states and control variables are obtained and depicted during orbit transfer. The obtained results demonstrate considerable convergence as well as sufficient accuracy.

Порівняльна ефективність дросельованих і релейних ракетних двигунів для навколоземних перельотів з малою тягою

Within the framework of solving the Mayer problem of optimal control of a flight from an elongated elliptical orbit to a geostationary one with a maximum payload at a given initial mass of the low-thrust spacecraft and a fixed duration of the dynamic maneuver, a comparative analysis was made of the efficiency of throttled and relay thrust modes. Based on the data of the numerical solution of the corresponding twopoint boundary value problem, the expected gain was confirmed of the throttling mode over the relay mode in the case of practically interesting low-thrust near-Earth maneuvers. Also the numerical results confirmed the adequacy of the constructed fixed-power relay rocket engine mathematical model and made it possible to reveal a number of qualitative features of the control functions along the optimal transition trajectories.

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.

Linearization Methods for Problems of Optimizing Low-Thrust Spacecraft Trajectories: Verification Procedure

Several well-studied problems for optimizing low-thrust spacecraft trajectories are solved using the modified linearization method (MLM). The results are compared with the results obtained by other authors. On this basis, conclusions are drawn on the possible use of the modified linearization method for optimizing low-thrust spacecraft trajectories.

Analytical approach to spacecraft formation-flying with low-thrust relative spiral trajectories

This work addresses the growing need for an intuitive, systematic approach to low-thrust spacecraft formation flying by extending shape-based continuous thrust trajectory design methods to the relative motion of two spacecraft. There is widespread interest in distributed space systems for their low costs, broad capabilities, and high redundancy. This trend introduces a new challenge for trajectory design when combined with the increasing prevalence of low-thrust, high specific impulse electric propulsion systems. That challenge is met herein with a geometrically intuitive, semi-analytical solution to the low-thrust problem. Beginning with the equations of relative motion of two spacecraft, an unperturbed chief and a continuously-thrusting deputy, a thrust profile is constructed which transforms the equations into a form that is solved analytically. The resulting relative trajectories are the family of sinusoidal spirals, which provide diversity for design and optimization based upon a single thrust parameter. Closed-form expressions are derived for the trajectory shape and time-of-flight for two prescribed relative velocity behaviors, and used to develop a novel patched-spirals trajectory design and optimization method. The example problem of a servicer spacecraft establishing and reconfiguring a formation around a target in geostationary earth orbit is used to demonstrate the application of the patched spirals technique as well as the advantages of the relative spiral trajectories over impulsive maneuvers. The sensitivity of the trajectory solutions to deviations from the underlying assumptions, uncertainties in the state, and errors in thrust are studied through high-fidelity simulation.

Invariant manifolds around artificial equilibrium points for low-thrust propulsion spacecraft

Low-thrust propulsion is incorporated into circular restricted three-body problem to balance the gravitational and centrifugal forces, and then artificial equilibrium points can be generated. The linear dynamics indicates that there are stable and unstable artificial equilibrium points. Around the unstable artificial equilibrium points, there are center and hyperbolic invariant manifolds. In this work, invariant manifolds around artificial equilibrium points are expressed as formal series of amplitudes corresponding to hyperbolic and center dynamics, and high-order series solutions are constructed up to an arbitrary order. By taking advantage of the series expansions constructed, the motions around unstable artificial equilibrium points can be parameterized. In order to check the validity, the practical convergence of series solutions truncated at different orders is considered. Finally, series expansions of invariant manifolds are applied to designing transfer trajectories from the primary to periodic orbits around artificial equilibrium points which are located inside $L_{1}$ and beyond $L_{2}$ points.

A new method for optimization of low-thrust gravity-assist sequences

Recently missions like Hayabusa and Dawn have shown the relevance and benefits of low-thrust spacecraft concerning the exploration of our solar system. In general, the efficiency of low-thrust propulsion is one means of improving mission payload mass. At the same time, gravity-assist maneuvers can serve as mission enablers, as they have the capability to provide “free energy.” A combination of both, gravity-assist and low-thrust propulsion, has the potential to generally improve mission performance, i.e. planning and optimization of gravity-assist sequences for low-thrust missions is a desirable asset. Currently no established methods exist to include the gravity-assist partners as optimization variable for low-thrust missions. The present paper explains how gravity-assists are planned and optimized, including the gravity-assist partners, for high-thrust missions and discusses the possibility to transfer the established method, based on the Tisserand Criterion, to low-thrust missions. It is shown how the Tisserand Criterion needs to be adapted using a correction term for the low-thrust situation. It is explained why this necessary correction term excludes an a priori evaluation of sequences and therefore their planning and an alternate approach is proposed. Preliminary results of this method, by application of a Differential Evolution optimization algorithm, are presented and discussed, showing that the method is valid but can be improved. Two constraints on the search space are briefly presented for that aim.

Comparison of Solar Sail Spacecraft Flight and Low Thrust Spacecraft Flight to Potentially Hazardous Asteroid Orbit

This paper outlines the nonlinear flight simulation of a low thrust spacecraft and a solar sail spacecraft. The main purpose of the work is ballistic comparison of solar sail spacecraft flight and low thrust spacecraft flight. The mathematical motion model for such types of spacecraft is formulated and described. The work considers the mathematical motion models within the heliocentric system of coordinates. Motion simulation process is the best way to assess the reasonableness of the mathematical model. On the basis of the formulated mathematical models, a special software for nonlinear flight simulation is developed. In particular, the flight of the low thrust spacecraft from Earth's orbit to the potentially hazardous asteroid is simulated. The findings of the work consist of heliocentric trajectories, time dependency graphs of flight parameters, the values of flight duration and the value of propellant consumption.

Influence of the Shaded Areas on the Low Thrust Spacecraft Flight Control in the Earth-moon system

The ballistic optimization of the L1-L2 and L2-L1 missions in the Earth-Moon system using electric propulsion is considered. The motion control is studied using the Fedorenko's successful linearization method to estimate the derivatives and the gradient method to optimize the control laws. The transfers are designed in frames of the restricted circular three-body problem. The mathematical model of the missions is described within the barycentric system of coordinates. The perturbation from the Earth, the Moon and the Sun are taking into account. The impact factor of the shaded areas, produced by the Earth and the Moon, is also accounted for. As for the results, the optimization of the optimal control laws and corresponding trajectories were obtained.

Simulation of Low Thrust Spacecraft Guided Motion for a Number of Missions in the Solar System

Models of spacecraft guided motion in the gravitational fields of different configurations were developed. Parameterization of the optimal control laws for all the trajectory segments of the spacecraft moving in the N-body gravitational field was realized. The dependence patterns of control laws on mission purposes were determined. The simulation of the number of missions in the Solar system was performed: low Earth orbit – L1 and L2 of the Earth-Moon and the Sun-Earth systems, L1 – L2 and L2 – L1 of the Earth-Moon and the Sun-Earth systems, low Earth orbit – asteroid Eros, low Earth orbit – Venus, low Earth orbit – Mars. Moreover, the Venus gravity assist maneuver was considered.

The Design of Low Thrust Engine Spacecraft for Near-earth Asteroid Exploration

This paper discusses the application of computer-assisted design systems to development of low thrust spacecraft intended to take an exploratory flight to potentially hazardous asteroids. The design model of the low thrust spacecraft that is created with computer-aided design systems is described. The mathematical motion model within the heliocentric system of coordinates for such type of spacecraft is considered and used for a motion simulation session. The simulation of flight to the potentially hazardous asteroid is performed with the aid of special complex software that is developed for the purpose of the work. The results of the paper consist of a detailed three-dimensional model of the low thrust spacecraft, heliocentric trajectory of the spacecraft, the values of flight duration and propellant consumption during exploratory flight.

Linearization methods for optimizing the low thrust spacecraft trajectory: Theoretical aspects

The theoretical aspects of the modified linearization method, which makes it possible to solve a wide class of nonlinear problems on optimizing low-thrust spacecraft trajectories (V. V. Efanov et al., 2009; V. V. Khartov et al., 2010) are examined. The main modifications of the linearization method are connected with its refinement for optimizing the main dynamic systems and design parameters of the spacecraft.