Low-thrust Orbit Transfer Research Articles

Multiple-arc optimization of low-thrust Earth–Moon orbit transfers leveraging implicit costate transformation

This work focuses on minimum-time low-thrust orbit transfers from a prescribed low Earth orbit to a specified low lunar orbit. The well-established indirect formulation of minimum-time orbit transfers is extended to a multibody dynamical framework, with initial and final orbits around two distinct primaries. To do this, different representations, useful for describing orbit dynamics, are introduced, i.e., modified equinoctial elements (MEE) and Cartesian coordinates (CC). Use of two sets of MEE, relative to either Earth or Moon, allows simple writing of the boundary conditions about the two celestial bodies, but requires the formulation of a multiple-arc trajectory optimization problem, including two legs: (a) geocentric leg and (b) selenocentric leg. In the numerical solution process, the transition between the two MEE representations uses CC, which play the role of convenient intermediate, matching variables. The multiple-arc formulation at hand leads to identifying a set of intermediate necessary conditions for optimality, at the transition between the two legs. This research proves that a closed-form solution to these intermediate conditions exists, leveraging implicit costate transformation. As a result, the parameter set for an indirect algorithm retains the reduced size of the typical set associated with a single-arc optimization problem. The indirect heuristic technique, based on the joint use of the necessary conditions and a heuristic algorithm (i.e., differential evolution in this study) is proposed as the numerical solution method, together with the definition of a layered fitness function, aimed at facilitating convergence. The minimum-time trajectory of interest is sought in a high-fidelity dynamical framework, with the use of planetary ephemeris and the inclusion of the simultaneous gravitational action of Sun, Earth, and Moon, along the entire transfer path. The numerical results unequivocally prove that the approach developed in this research is effective for determining minimum-time low-thrust Earth–Moon orbit transfers.

D Matrix for Keplerian State Error Propagation

The D matrix is an error state transition matrix (STM) for Keplerian motion in cylindrical coordinates employing time as a state variable and reference true or eccentric anomaly as an independent variable. It has been only partially derived in previous articles and applied to different astrodynamics problems such as optimal collision avoidance and Lambert targeting. A rigorous and complete derivation of the full matrix in analytical form is presented. In addition, it is shown that the different terms of the proposed STM can lead to the definition of new variational equations and optimal steering laws applicable to impulsive and low-thrust orbit transfer optimization as well as reachability analysis. Finally, the effectiveness of the proposed STM is evaluated against the well-known Yamanaka–Ankersen STM showing more than one order of magnitude error reduction associated to the semimajor axis and eccentricity of the linearly propagated relative state.

Optimal Low-Thrust Earth Orbit Transfers with Eclipses Using Indirect Heuristic Approaches

Low-thrust propulsion allows substantial propellant saving if compared to high-thrust systems. However, multirevolution orbit transfers are affected by the unavailability of electrical power along eclipse arcs. This research addresses the identification of minimum-time low-thrust Earth orbit transfers with spacecraft eclipsing. Two different indirect heuristic approaches, based on the use of the analytical conditions for optimality, in conjunction with stochastic fractal search, are applied: 1) regularization through the hyperbolic tangent function and 2) multi-arc formulation of the problem. Both of them require no averaging, and the respective advantages and shortcomings are identified. Approach 2 leads to an extended set of multipoint conditions for optimality, which are shown to be solvable sequentially in the numerical solution process. Implicit costate transformation is proven to be a key ingredient for the purpose of obtaining closed-form solutions of the jump relations that hold for the adjoint variables at light/eclipse transitions. Some illustrative examples prove effectiveness of the two indirect heuristic approaches 1 and 2 in finding minimum-time low-thrust orbit transfers between either two coplanar or two noncoplanar Earth orbits. Approach 2 proves to be superior in terms of both computational efficiency and accuracy of the numerical results.

Second-Order Averaging of Low-Thrust Transfers in Fixed Time

An algorithm for the second-order approximation of fixed-time, low-thrust orbital transfers and generation of Bacon plots is proposed. After averaging the extremal flow of the optimal-control Hamiltonian, a one-parameter family of solutions of a reduced-order, two-point boundary value problems is computed by means of a differential-continuation scheme. Sensitivities of the shooting function are then used in conjunction with an ad hoc near-identity transformation between averaged and osculating variables to achieve an accurate solution for all longitudes of the departure and arrival orbits. Hence, a single simplified shooting problem has to be solved to approximate the solution for any combination of departure and arrival dates and to draw a Bacon plot for an arbitrary launch window. Both the averaged flow and the near-identity transformation are efficiently evaluated via the fast-Fourier transform algorithm, yielding a fully numerical procedure.

Optimal Design and Tracking Control of Low-Thrust Orbit Transfers Between Planar and Vertical Lyapunov Orbits

Optimal Design and Tracking Control of Low-Thrust Orbit Transfers Between Planar and Vertical Lyapunov Orbits

Low-Thrust Transfer Orbit Optimization Using Sequential Convex Programming and Adaptive Mesh Refinement

In this paper, a mesh-adaptive Hermite–Simpson sequential convex programming (SCP) method is proposed for solving low-thrust orbit transfer problems efficiently and accurately. First, we develop a Hermite–Simpson convex programming method that utilizes quadratic polynomial control interpolation to improve discretization accuracy. Then, a mesh-adaptive Hermite–Simpson SCP method is proposed by incorporating an improved adaptive mesh refinement method into the Hermite–Simpson SCP method for solving optimal control accurately and efficiently. The proposed method is applied to two-dimensional time-optimal and three-dimensional fuel-optimal low-thrust orbit transfer problems to show its features, and its performance is compared against several existing methods. Numerical simulations show that the proposed method achieves more accurate solutions while being computationally more efficient compared to the trapezoidal SCP method. In comparison to the GPOPS-II software, the proposed method produces solutions of comparable accuracy but is more computationally efficient.

Robust estimation of distribution algorithms via fitness landscape analysis for optimal low-thrust orbital maneuvers

One particular kind of evolutionary algorithms known as Estimation of Distribution Algorithms (EDAs) has gained the attention of the aerospace industry for its ability to solve nonlinear and complicated problems, particularly in the optimization of space trajectories during on-orbit operations of satellites. This article describes an effective method for optimizing the trajectory of a spacecraft using an evolutionary approach based on EDAs, incorporated with fitness landscape analysis (FLA). The approach utilizes flexible operators that are paired with seeding and selection mechanisms of EDAs. Initially, the orbit transfer problem is mathematically modeled and the objectives and constraints are identified. The landscape feature of the search space is analyzed via the dispersion metric to measure the modality and ruggedness of the search domain. The obtained information are used as feedback in developing adaptive operators for truncation factor and constraints separation threshold of the employed EDA. A framework for spacecraft trajectory optimization has been presented where the dispersion value for a space mission is estimated using a k-nearest neighbors (k-NN) algorithm. The suggested method is used to solve several problems related to low-thrust orbit transfer of satellites in Earth’s orbit. Results demonstrate that the suggested framework for trajectory design and optimization of space transfers is effective enough to offer fuel-efficient and energy-efficient maneuvers for different thrust levels of the propulsion system. Moreover, the performance of the proposed approach is evaluated against non-adaptive EDA and other advanced evolutionary algorithms. The obtained results certify that the proposed adaptive evolutionary approach is superior in identifying feasible minimum-fuel and minimum-energy transfer trajectories.

Multi-objective low-thrust spacecraft trajectory design using reachability analysis

One of the fundamental problems in spacecraft trajectory design is finding the optimal transfer trajectory that minimizes the propellant consumption and transfer time simultaneously. We formulate this as a multi-objective optimal control (MOC) problem that involves optimizing over the initial or final state, subject to state constraints. Drawing on recent developments in reachability analysis subject to state constraints, we show that the proposed MOC problem can be stated as an optimization problem subject to a constraint that involves the sub-level set of the viscosity solution of a quasi-variational inequality. We then generalize this approach to account for more general optimal control problems in Bolza form. We relate these problems to the Pareto front of the developed multi-objective programs. The proposed approach is demonstrated on two low-thrust orbital transfer problems around a rotating asteroid.

Engineering Optimization Method of Orbit Transfer Strategy for All-electric Propulsion Satellites

Nowadays, all-electric propulsion satellites have shown very good performance in missions such as geostationary communication and deep space exploration, and have been widely concerned by major aerospace companies around the world. Because the excellent performance of the electric thruster can enable the satellite to carry more payloads. While the use of electric thrusters for orbit transfer saves fuel consumption significantly, it also leads to the problem of long orbit transfer time. To solve this problem, a simplified low-thrust orbit transfer strategy is designed, which is simple in form, easy to store, while having few parameters to be optimized. In order to optimize the simplified transfer strategy, a bidirectional stochastic gradient descent method suitable for on-board calculation is proposed, with a simple algorithm process and small computation. It can be known from actual calculations that after 600 times of orbit extrapolation, it can converge to the optimal solution. Finally, we can draw a conclusion that the algorithm is feasible and effective by numerical simulation.

Resonant Control of Satellite Orbits

Low-thrust orbit control has been widely studied, mostly by using direct or indirect optimization to detect time-optimal and fuel-optimal orbit transfers. This paper proposes an alternative approach for designing efficient low-thrust orbit transfers. The main idea is to create an artificial resonance between the period of the control accelerations and a characteristic period of the orbital dynamics. To that end, Gauss’s variational equations are written using a Fourier series expansion in the mean anomaly. The secular changes of the orbital elements are determined based on averaging theory. It is shown that an artificially created low-thrust resonance can save fuel compared with piecewise-constant thrust. A general resonant-control candidate for changing all mean orbital elements efficiently is proposed. Four simple decoupling control laws are designed for changing the semimajor axis, eccentricity, inclination, and right ascension of the ascending node separately. In addition, periodic corrections transforming between mean elements and osculating elements are derived. The proposed decoupling control laws are applied to a formation-keeping problem, illustrating the potential merit of the new control law compared with a fixed-thrust-magnitude feedback control.

Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach

The optimization of low-thrust, multirevolution orbit-transfer trajectories is often regarded as a difficult problem in modern astrodynamics. In this paper, a flexible and computationally efficient approach is presented for the optimization of low-thrust orbit transfers under eclipse constraints. The proposed approach leverages a new dynamic model of the orbital motion and a Lyapunov-based initial-guess generation scheme that is very easy to tune. A multi-objective, single-phase formulation of the optimal control problem is devised, which provides a convenient way to trade off fuel consumption and time of flight. A distinctive feature of such a formulation is that it requires no prior information about the structure of the optimal solution. Simulation results for two benchmark orbit-transfer scenarios indicate that minimum-time, minimum-fuel, and mixed time/fuel-optimal instances of the control problem can be readily solved via direct collocation, while incurring a significantly lower computational demand with respect to existing techniques.

Analytical solutions for low-thrust orbit transfers

This paper presents analytical solutions for the estimation of the Delta V cost of the transfer of a spacecraft subject to a low-thrust action. The equations represent an extension of solutions already available in the literature. Moreover, the paper presents novel analytical solutions for low-thrust transfers under the effect of the second-order zonal harmonics of the Earth’s gravitational potential. In particular, the paper is divided into two parts. The first part presents analytical expressions for the Delta V cost of transfers. All analytical equations were validated through numerical integration of the dynamics of the spacecraft. The second part of the paper introduces new analytical equations for low-thrust transfers between circular inclined orbits with different values of the right ascension of the ascending node, under the effect of the second-order zonal harmonic of the Earth’s gravitational potential. Both in the first and second parts, analytic solutions for the variation with time of the orbital elements during the transfer are presented. The proposed equations are applicable to low-thrust transfer realised through a long spiral trajectory.

Quasi-time-optimal steering law for low-thrust orbit transfer considering angular momentum and torque constraints

In this study, a quasi-time-optimal steering law for low-thrust orbit transfer is investigated. In the proposed steering law, the satellite attitude is controlled to suppress its error with respect to the ideal thruster direction under the orthogonality constraint between the Sun direction and the solar array driving mechanism's rotational axis as well as the angular momentum and torque constraints. All these constraints are satisfied by introducing a structure similar to Baumgarte's stabilization method and by formulating a quadratically constrained quadratic programming problem which can be analytically solved to obtain the control torque. Several numerical simulations show that the transfer time difference from the ideal unconstrained transfer is less than a few percent.

Differential correction method in low-thrust orbit transfers near the collinear libration points

Differential correction method in low-thrust orbit transfers near the collinear libration points

Bounding Homotopy Method for Minimum-Time Low-Thrust Transfer in the Circular Restricted Three-Body Problem

Solving minimum-time low-thrust orbital transfer problems in the three-body problem by indirect methods is an extremely difficult task, which is mainly due to the small convergence domain of the optimal solution and the highly nonlinear nature associated with the three-body problem. Homotopy methods, the principle of which is to embed a given problem into a family of problems parameterized by a homotopic parameter, have been utilized to address this difficulty. However, it is not guaranteed that the optimal solution of the original problem can be obtained by most of the existing homotopy methods. In this paper, a new bounding homotopy method is proposed, by which the continuous homotopy path can be constructed and the optimal solution of the original problem is guaranteed to be found. In the parameter bounding homotopy method, an initialized problem with much higher thrust is constructed and a state-of-the-art parameter bounding homotopy approach is utilized to connect separated homotopy branches outside the predefined domain of the homotopic parameter. Furthermore, multiple optimal solutions of the original problem can be obtained if the homotopic approach continues after the first solution, among which the best solution can be figured out. Finally, numerical solutions of minimum-time low-thrust orbital transfers from GEO to Moon orbit and from GTO to halo orbit in the circular restricted three-body problem are provided to demonstrate the effectiveness of the homotopy method.

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.

Orbit Control Techniques for Space Debris Removal Missions Using Electric Propulsion

Electric propulsion is currently seen as a key enabling technology for space debris removal missions aimed at deorbiting multiple debris targets. This paper develops an autonomous onboard orbit control strategy tailored to these missions. The control problem is divided into four stages, involving a sequence of low-thrust orbital transfer and rendezvous maneuvers. A feedback control law is derived for each maneuvering stage, by exploiting Lyapunov-based and model predictive control techniques. The proposed design is able to account for mission-specific performance and safety requirements, while satisfying on–off constraints inherent to the propulsion technology. Simulation case studies of a multidebris removal mission demonstrate the effectiveness of the proposed control strategy, and support the viability of electric propulsion for such type of missions.

A new fractional homotopy method for solving nonlinear optimal control problems

The homotopy methods have long served as powerful tools in solving nonlinear optimal control problems, particularly for which the solutions are highly sensitive to the unknown initial conditions. The principle of homotopy methods is that a homotopic parameter is embedded into the formulations of the optimal control problems, and the original problem can be solved by tracing the optimal solutions of the embedded problems. The existing homotopy methods typically introduce the homotopic parameter into the time variable, the necessary conditions, the performance index or the rightside of the differential equations. In this paper, a new fractional homotopy method is presented, the homotopic parameter of which is embedded into the derivative of the differential equations. By using the proposed method, the optimal solution of the target homotopy problem can be found by solving a series of fractional two-point-boundary-value-problems. Numerical demonstrations in a nonlinear optimal control problem and a three-dimensional minimum-time low-thrust orbital transfer problem are presented to illustrate the applications of the method.

On-orbit servicing of geosynchronous satellites based on low-thrust transfers considering perturbations

The goal of this paper is to investigate the on-orbit servicing (OOS) of geosynchronous satellites based on low-thrust transfers while considering the different demands of the satellites and the effects of perturbations. One or more servicing spacecraft start from a parking orbit and service the satellites in sequence via low-thrust orbit transfers, and return to the parking orbit in the end. The different demands lead to different service requirements of time and mass, while the perturbations affect the control solution of servicing spacecraft during each transfer, and induce orbit drifts of uncontrolled target satellites during the mission. An analytical method that can efficiently calculate the fuel consumption of low-thrust transfer is proposed, and methods for evaluating the orbit drifts of uncontrolled satellites are presented. The impact of perturbations on OOS mission is investigated in detail, showing that non-negligible factors include lunisolar attraction and tessral harmonic. Based on the dynamic models, a string-based simulated annealing algorithm is adopted to schedule the OOS mission. The efficiency of the proposed dynamic models and optimization method are verified through ample simulations. Results show that it is much more efficient and reasonable to take orbit drift calculations as corrections on the scheduling result assuming no orbit drifts. The proposed models and methods make it possible to schedule the OOS mission with low-thrust transfers, while achieving high accuracy under the effects of J2 perturbation, lunisolar attracion, tesseral harmonic and solar radiation pressure.

Koopman Reduced Order Control for Three Body Problem

In this paper, we use a Circle Restricted Three-Body Problem (CRTBP) to simulate the motion of a satellite. Then we reformulate this problem with the controller into the description using Koopman eigenfunction. Although the original dynamical system is nonlinear, the Koopman eigenfunction behaves linearly. Choosing Jacobi integral as the Koopman eigenfunction and using the zero velocity curve as the reference for control, we are allowed to combine well-developed Linear Quadratic Regulator (LQR) controller to design a nonlinear controller. Using this approach, we design the low thrust orbit transfer strategy for the satellite flying from the earth to the moon or from the earth to the sun.