Orbit Transfer Problem Research Articles

Spectral Shifted Stabilized Continuation for Indirect Optimal Control

Continuation methods provide a solution to the initial guess problem that numerical root solvers and shooting methods often face by iteratively transforming an auxiliary solution into a desired solution through differential corrections. While seemingly straightforward, designing an automated continuation strategy that starts at the auxiliary problem and successfully terminates at the desired problem with minimal user intervention is often challenging especially for optimal trajectory-planning problems with nonlinear motion models subject to state/path and control constraints. One scenario where the auxiliary problem does not connect to the desired problem occurs when the intermediate problems are ill posed (i.e., infeasible). This paper presents a two-layered Jacobian conditioned stabilized continuation algorithm that circumvents these infeasible zones along a user-defined continuation path with little designer intervention. The efficacy of this approach is evaluated in the context of several prototype problems including quadratic root-solving, optimal path planning with a Dubins model, an optimal orbit transfer problem, and an optimal high-speed vehicle trajectory generation problem.

Orbital Transfer Problem on the Central Manifold of Libration Points

Orbital Transfer Problem on the Central Manifold of Libration Points

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.

Adaptive Estimation of Distribution Algorithms for Low-Thrust Trajectory Optimization

A direct adaptive scheme is presented as an alternative approach for minimum-fuel low-thrust trajectory design in non-coplanar orbit transfers using fitness landscape analysis (FLA). The spacecraft dynamics is modeled with respect to modified equinoctial elements by considering orbital perturbations. Taking into account the timings of thrust arcs, the discretization nodes for thrust profile, and the solution of multi-impulse orbit transfer, a constrained continuous optimization problem is formed for low-thrust orbital maneuvers. An adaptive method within the framework of the estimation of distribution algorithms (EDA) is proposed, which aims at conserving the feasibility of the solutions within the search process. Several problem identifiers for low-thrust trajectory optimization are introduced, and the complexity of the solution domain is analyzed by evaluating the landscape feature of the search space via FLA. Two adaptive operators are proposed, which control the search process based on the need for exploration and exploitation of the search domain to achieve optimal transfers. The adaptive operators are implemented in the presented EDA and several perturbed and nonperturbed orbit transfer problems are solved. The results confirm the effectiveness and reliability of the proposed approach in finding optimal low-thrust transfer trajectories.

Correction Strategy of Online Midcourse Guidance for High-Speed Gliding Target Interceptor

During the interception of a high-speed gliding vehicle (HGV), the predicted intercept point (PIP) is updated many times, so the interceptor must continuously adjust its trajectory. In this paper, we propose an online guidance correction algorithm for the interceptor, which can effectively meet the position constraints and energy management of the guidance process. Firstly, we simplify the interceptor’s maneuvering problem to a two-body orbital transfer problem. The Lambert orbit transfer method is used to enable the interceptor to maneuver towards the target’s PIP. Secondly, by comparing the target’s high probability existence area (THPEA) with the interceptor-interceptable area (IIA), the engine start threshold is set in stages to avoid frequent redundancy maneuvers caused by frequent engine starting during missile correction. This approach achieves energy management. Simulation results show that our midcourse guidance strategy is effective.

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.

A mesh refinement method for low‐thrust orbit transfer problems

AbstractThis paper researches the fuel‐optimal geocentric orbit transfer problem using a low‐thrust, electric propulsion. First, the transfer trajectory is divided into N burn‐coast arcs via introducing (N−1) intermediate orbits between the initial and the target orbits. By restricting the time duration of each burn, the changes in orbital states caused by one time of ignition can be represented by a discrete velocity impulse; therefore, a considerable large N, a great many optimization variables, and a months and revolutions long trajectory will be calculated to solve the optimization problem. By proposing a new transfer coordinate system, the performance index is analytically calculated, and strict constraints derived from the continuous condition between adjacent orbits are removed from the mathematical optimization model. Meanwhile, an improved particle swarm optimization (PSO) method and a mesh refinement method are proposed to address the optimization problem. The mesh refinement algorithm begins with a simple bi‐impulse transfer problem, automatically increases/decreases N, and finally converges to the optimal solution. Numerical simulation shows the efficiency of the method with some classical and representative scenarios.

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.

Impact of Using Analytic Derivatives In Optimization For N-Impulse Orbit Transfer Problems

Impact of Using Analytic Derivatives In Optimization For N-Impulse Orbit Transfer Problems

Physics-Informed Neural Networks for Optimal Planar Orbit Transfers

This paper presents a novel framework, combining the indirect method and Physics-Informed Neural Networks (PINNs), to learn optimal control actions for a series of optimal planar orbit transfer problems. According to the indirect method, the optimal control is retrieved by directly applying the Pontryagin minimum principle, which provides the first-order necessary optimality conditions. The necessary conditions result in a two-point boundary value problem (TPBVP) in the state–costate pair, constituting a system of ordinary differential equations, representing the physics constraints of the problem. More precisely, the goal is to model a neural network (NN) representation of the state–costate pair for which the residuals of the TPVBP are as close to zero as possible. This is done using PINNs, which are particular NNs where the training is driven by the problem’s physics constraints. A particular PINN method will be used, named Extreme Theory of Functional Connections (X-TFC), which is a synergy of the classic PINN and the Theory of Functional Connections. With X-TFC, the TPBVP’s boundary conditions are analytically satisfied. This avoids having unbalanced gradients during the network training. The results show the feasibility of employing PINNs to tackle this class of optimal control problems for space applications.

Graceful Transitions through Maximal Persistence of Behavior

Problems of locomotion typically encounter switching between gaits. In many applications, it is desirable to make such transitions between modes or gaits inconspicuous and graceful. This is achieved by keeping the typical behavior of the system as persistent as possible. It is shown that this problem can be defined in a more abstract sense, and therefore generalized to a new class of problems, called “maximum persistence of behavior.”In this presentation, I will give a precise definition of this concept, for signals and for systems, and illustrate some applications from thermodynamics to signal study and robotics. I will also introduce the dual problem of filtering in the context of “maximal persistence of behavior.” The results stem from joint work with Deryck Yeung (Trinity University, San Antonio, TX), Basit Memon (Habib University, Karachi, PK), and Vishal Murali (now at NVIDIA) while at Georgia Tech.The paradigm starts by defining typical behaviors in the framework initiated by J.C. Willems. Gluing two typical signals or typical system behaviors requires connections (raccordations) that belong to a larger class of behaviors, but are locally close to the typical behaviors. Solutions with persistent behavior are then sought via optimization for a kernel or an image problem. This solution encapsulates the controllability problem of Willems, morphing theory in image analysis, object shaping, quasi-stationary transitions from thermodynamics, and orbit transfer problems, and can be characterized in a geometric way. In the robotics framework, obvious applications are in gait transitions for biomimetic robots as when switching from walking (e.g., while inspecting) to running (evasion), or a gait transition necessitated by a change in terrain (solid, muddy, or granular). In particular, I will discuss a method of smoothly connecting periodic orbits of systems. We first deal with the case when the periodic signals to be connected are of the same period and then move on to discuss the case of different periods, requiring in addition a frequency warping.

A quasi-linear local variational iteration method for orbit transfer problems

A novel Quasi-linear Local Variational Iteration Method (QLVIM) for solving two-point boundary value problems (TPBVPs) of strongly nonlinear systems is proposed. With quasi-linearization technique, the multidimensional nonlinear TPBVP is transformed into a series of iterative linear TPBVPs, and then into initial value problems (IVPs) which appear in pairs. Then these IVPs are solved by Local Variational Iteration Method (LVIM). Combining the rapid convergence of quasi-linearization with the high computational efficiency of LVIM, the proposed QLVIM can solve various orbit transfer problems in aerospace engineering efficiently and accurately. The high-performance of this method is verified in solving perturbed Lambert's problems and a circular restricted three-body orbit transfer problem. Comparisons with various methods show that the QLVIM has the advantages of high convergence speed, low computation cost and longer solvable time span. Rather than a refined initial guess of starting velocity, the QLVIM only needs a rough initial guess of the trajectory. The results also prove that the QLVIM is capable of solving long-time-span orbit transfer problem. A theoretical proof of its quadratic convergence is given at last.

Second-Order Conditions for Fuel-Optimal Control Problems with Variable Endpoints

Second-order optimality conditions for fuel-optimal control problems with both endpoints on manifolds are studied in this paper. Each candidate extremal from Pontryagin’s maximum principle is embedded into a parameterized family of extremals, and two no-fold conditions for the projection of the parameterized family onto state space to be a diffeomorphism are devised. By the theory of field of extremals, the no-fold conditions are related to the classical second-order conditions for the absence of conjugate points or focal points. Thus, the no-fold conditions are sufficient to ensure the candidate extremal to be locally optimal if the initial state is fixed. When the initial state is not fixed but lies on a manifold, an additional second-order optimality condition involving the geometry of the initial constraint manifold is established. A numerical procedure for testing all the second-order conditions is presented, allowing to check the optimum of each candidate extremal. Finally, three fuel-optimal orbital transfer problems in the Earth–moon–spacecraft system are computed, demonstrating and verifying the developments of the paper.

Successive Chebyshev pseudospectral convex optimization method for nonlinear optimal control problems

AbstractThis article aims at proposing a successive Chebyshev pseudospectral convex optimization method for solving general nonlinear optimal control problems (OCPs). First, Chebyshev pseudospectral discrete scheme is used to discretize a general nonlinear OCP. At the same time, a convex subproblem is formulated by using the first‐order Taylor expansion to convexify the discretized nonlinear dynamic constraints. Second, a trust‐region penalty term is added to the performance index of the subproblem, and a successive convex optimization algorithm is proposed to solve the subproblem iteratively. Noted that the trust‐region penalty parameters can be adjusted according to the linearization error in iterative process, which improves convergence rate. Third, the Karush–Kuhn–Tucker conditions of the subproblem are derived, and furthermore, a proof is given to show that the algorithm will iteratively converge to the subproblem. Additionally, the global convergence of the algorithm is analyzed and proved, which is based on three key lemmas. Finally, the orbit transfer problem of spacecraft is used to test the performance of the proposed method. The simulation results demonstrate the optimal control is bang‐bang form, which is consistent with the result of theoretical proof. Also, the algorithm is of efficiency, fast convergence rate, and high accuracy. Therefore, the proposed method provides a new approach for solving nonlinear OCPs online and has great potential in engineering practice.

Minimum-Time Earth-to-Mars Interplanetary Orbit Transfer Using Adaptive Gaussian Quadrature Collocation

The problem of minimum-time, low-thrust, Earth-to-Mars interplanetary orbital trajectory optimization is considered. The minimum-time orbital transfer problem is modeled as a four-phase optimal control problem where the four phases correspond to planetary alignment, Earth escape, heliocentric transfer, and Mars capture. The four-phase optimal control problem is then solved using a direct collocation adaptive Gaussian quadrature collocation method. The following three models are used in the study: 1) circular planetary motion, 2) elliptic planetary motion, and 3) elliptic planetary motion with gravity perturbations, where the transfer begins in a geostationary orbit and terminates in a Mars-stationary orbit. Results for all three cases are provided, and one particular case is studied in detail to show the key features of the optimal solutions. Using the particular value thrust specific force of , it was found that the minimum times for cases 1, 2, and 3 are, respectively, 215, 196, and 198 d with departure dates, respectively, of 1 July 2020, 30 June 2020, and 28 June 2020. Finally, the problem formulation developed in this study is compared against prior work on an Earth-to-Mars interplanetary orbit transfer where it is found that the results of this research show significant improvement in transfer time relative to the prior work.

Piecewise attitude tracking control of a gravity gradient microsatellite for coplanar orbital transfer

AbstractThe gravity gradient microsatellite applied to the exploration of space environment is subject to the problem of orbital transfer between two orbits. The contributions of this paper mainly include two aspects: (1) a segmented optimal trajectory and (2) the piecewise terminal tracking control method. In order to achieve orbital transfer between two coplanar orbits, an optimal controller is designed to obtain a segmented optimal trajectory and then it becomes a problem of attitude tracking in different segments. Compared with traditional finite‐time control methods, the adaptive sliding mode control method based on a finite‐time tracking error function is proposed, and the stability and robustness of the system can be guaranteed. A numerical case between two coplanar orbits is studied to validate the effectiveness of the control method.

Optimal Multi-Target Overflight Using Ground-Track Adjustment

This study proposes a near-optimal, impulsive control method for overflying ground targets. A set of impulses, which adjusts the satellite ground trajectory to enable the overflight, is derived. The proposed impulsive solution is expressed in closed-form which only consists of orbital revolution numbers and target overflight errors. It allows not only simplifying the search for a minimum delta-v solution but also facilitating its extension to the multiple-target overflight problem. It is also provided in this paper an insight connecting the overflight problem to a relative orbit transfer problem controlling a phase angle or mean anomaly. Numerical examples for single-, two-, and three-target overflight verify that the controlled trajectory with the proposed impulsive solution successfully overflies the designated targets within a prescribed time. It is also demonstrated that the total delta-v required for the proposed method is near-optimal based on the performance comparison with a numerical optimization-based method which employs a very large set of initial guesses.

A Newly Constant-Thrust Control Method Based on Parameter Estimation

Finite-thrust control technology is widely applied to space missions such as orbital transfer and rendezvous. However, traditional low-thrust method requires time-varying acceleration for achieving transfer and rendezvous. In this study, a newly piecewise constant thrust method is proposed for orbital transfer problems considering J 2 perturbation. To rapidly obtain the solution, UKF parameter estimation method is adopted, which transform the original problem into a parameter estimation problem. Numerical simulations are conducted to demonstrate the validation of the proposed method. It proves to be a practical choice for engineering applications.

Multi-fidelity and multi-objective optimization of low-thrust transfers with control strategy for all-electric geostationary satellites

To reduce the transfer time and radiation damage of all-electric geostationary satellites, this article develops a metamodel based multi-objective optimization scheme for low-thrust geostationary Earth orbit transfers. A multi-stage control strategy is employed to convert the optimal Earth orbital transfer problem into a parameterized multi-objective optimization problem with minimum total transfer time and proton displacement dose objectives. Based on the notion of multi-model fusion, the multi-fidelity system dynamics and radiation damage models are established to reduce the optimization cost. Furthermore, a novel adaptive Co-Kriging based multi-objective optimization method is proposed to effectively solve the multi-objective optimization problem with multi-fidelity models. In this approach, Co-Kriging metamodels are constructed using the multi-fidelity data to approximate the time-consuming low-thrust transfer models for optimization. And the Co-Kriging metamodels are adaptively refined by the infill samples to explore the Pareto frontier efficiently. Finally, a real-world all-electric geostationary satellite transfer example is investigated. The results show that a number of non-dominated low-thrust trajectories are successfully obtained with limited computational budget. Moreover, the total transfer time and displacement damage dose are significantly reduced after optimization, which illustrates the effectiveness of the metamodel based multi-objective low-thrust transfer optimization scheme.

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.