Space Trajectory Design Research Articles

Chaotic Artificial Algae Algorithm for Solving Global Optimization With Real-World Space Trajectory Design Problems

AbstractThe artificial algae algorithm (AAA) is a recently introduced metaheuristic algorithm inspired by the behavior and characteristics of microalgae. Like other metaheuristic algorithms, AAA faces challenges such as local optima and premature convergence. Various strategies to address these issues and enhance the performance of the algorithm have been proposed in the literature. These include levy flight, local search, variable search, intelligent search, multi-agent systems, and quantum behaviors. This paper introduces chaos theory as a strategy to improve AAA's performance. Chaotic maps are utilized to effectively balance exploration and exploitation, prevent premature convergence, and avoid local minima. Ten popular chaotic maps are employed to enhance AAA's performance, resulting in the chaotic artificial algae algorithm (CAAA). CAAA's performance is evaluated on thirty benchmark test functions, including unimodal, multimodal, and fixed dimension problems. The algorithm is also tested on three classical engineering problems and eight space trajectory design problems at the European Space Agency. A statistical analysis using the Friedman and Wilcoxon tests confirms that CAA demonstrates successful performance in optimization problems.

Convex Approach to Covariance Control with Application to Stochastic Low-Thrust Trajectory Optimization

This paper outlines a novel approach to the design of optimal space trajectories under significant uncertainty. Finite-horizon covariance control, i.e., the steering of a system from an initial probability distribution to a desired one at a prescribed time, is employed to plan an optimal nominal path along with a robust feedback controller that compensates for exogenous in-flight disturbances. A mindful convexification strategy is devised to recast the nonlinear covariance control problem as a deterministic convex optimization problem. The convexification is based on a convenient change of variables that allows for relaxing the covariance matrix discrete-time propagation into a set of semidefinite cone constraints. While featuring a larger feasible space, the relaxed problem shares the same optimal solution as the original one. The lossless property of the relaxation is analytically proven by leveraging optimal control theory and discussed on the basis of numerical experiments. Monte Carlo campaigns are carried out to validate the in-flight performance of the attained control policies.

LEO constellation design methodology for observing multi-targets

This paper is based on the second problem of the 8th China Space Trajectory Design Competition (CTOC8). The background is LEO constellation design strategy for monitoring discrete multi-targets with small satellite under J2 model. The difficulty is that the small satellite is equipped with low-cost cameras with limited coverage ability and the targets are distributed separately in a key area, which result in long revisit time or large number of satellites based on traditional design method. In this paper, a specific LEO constellation design method is proposed to cope with the problems. First, grid search and numerical method are performed to construct a database consisting of repeating ground track orbits. Then several orbits are carefully selected by pruning method to visit each target. Finally, repeating ground track constellation is constructed to meet the maximum revisit time constraint. The present method provides a systematic constellation design methodology of remote sensing observation with limited coverage ability, and demonstrates the resulting constellation can obtain rapid revisit frequency over discrete multi-targets with the least number of satellites.

Parallelization Impact on Many-Objective Optimization for Space Trajectory Design

Parallelization Impact on Many-Objective Optimization for Space Trajectory Design

Optimal design of near-Earth asteroid sample-return trajectories in the Sun–Earth–Moon system

In the 6th edition of the Chinese Space Trajectory Design Competition held in 2014, a near-Earth asteroid sample-return trajectory design problem was released, in which the motion of the spacecraft is modeled in multi-body dynamics, considering the gravitational forces of the Sun, Earth, and Moon. It is proposed that an electric-propulsion spacecraft initially parking in a circular 200-km-altitude low Earth orbit is expected to rendezvous with an asteroid and carry as much sample as possible back to the Earth in a 10-year time frame. The team from the Technology and Engineering Center for Space Utilization, Chinese Academy of Sciences has reported a solution with an asteroid sample mass of 328 tons, which is ranked first in the competition. In this article, we will present our design and optimization methods, primarily including overall analysis, target selection, escape from and capture by the Earth–Moon system, and optimization of impulsive and low-thrust trajectories that are modeled in multi-body dynamics. The orbital resonance concept and lunar gravity assists are considered key techniques employed for trajectory design. The reported solution, preliminarily revealing the feasibility of returning a hundreds-of-tons asteroid or asteroid sample, envisions future space missions relating to near-Earth asteroid exploration.

Survey of Direct Transcription for Low-Thrust Space Trajectory Optimization with Applications

Space trajectory design is usually addressed as an optimal control problem. Although it relies on the classic theory of optimal control, this branch possesses some peculiarities that led to the development of ad hoc techniques, which can be grouped into two categories: direct and indirect methods. This paper gives an overview of the principal techniques belonging to the direct methods. The technique known as “direct transcription and collocation” is illustrated by considering Hermite-Simpson, high-order Gauss-Lobatto, and pseudospectral methods. Practical examples are given, and several hints to improve efficiency and robustness are implemented.

Multi Agent Collaborative Search based on Tchebycheff decomposition

This paper presents a novel formulation of Multi Agent Collaborative Search, for multi-objective optimization, based on Tchebycheff decomposition. A population of agents combines heuristics that aim at exploring the search space both globally (social moves) and in a neighborhood of each agent (individualistic moves). In this novel formulation the selection process is based on a combination of Tchebycheff scalarization and Pareto dominance. Furthermore, while in the previous implementation, social actions were applied to the whole population of agents and individualistic actions only to an elite subpopulation, in this novel formulation this mechanism is inverted. The novel agent-based algorithm is tested at first on a standard benchmark of difficult problems and then on two specific problems in space trajectory design. Its performance is compared against a number of state-of-the-art multi-objective optimization algorithms. The results demonstrate that this novel agent-based search has better performance with respect to its predecessor in a number of cases and converges better than the other state-of-the-art algorithms with a better spreading of the solutions.

A hybrid, self-adjusting search algorithm for optimal space trajectory design

The aim of the present paper is to propose a hybrid, self adjusting, search algorithm for space trajectory optimization. By taking advantage of both direct and indirect methods, the present algorithm allows the finding of the optimal solution through the introduction of some new control parameters, whose number is smaller than that of the Lagrange multipliers, and whose range is bounded. Eventually, the optimal solution is determined by means of an iterative self-adjustment of the search domain occurring at “runtime”, without any correction by an external user. This new set of parameters can be found through a reduction process of the degrees of freedom, obtained through the transversality conditions before entering the search loop. Furthermore, such a process shows that Lagrange multipliers are subject to a deep symmetry mirroring the features of the state vector. The algorithm reliability and efficiency is assessed through some test cases, and by reproducing some optimal transfer trajectories: a full three-dimensional, minimum time Mars mission, an optimal transfer to Jupiter, and finally an injection into a circular Moon orbit.

Localization by nonlinear phase preparation and k‐space trajectory design

A technique is described to localize MR signals from a target volume using nonlinear pulsed magnetic fields and spatial encoding trajectories designed using local k-space theory. The concept of local k-space is outlined theoretically, and this principle is applied to simulated phantom and cardiac MRI data in the presence of surface and quadrupolar gradient coil phase modulation. Phantom and in vivo human brain images are obtained using a custom, high-performance quadrupolar gradient coil integrated with a whole-body 3-T MRI system to demonstrate target localization using three-dimensional T 2*-weighted spoiled gradient echo, two-dimensional segmented, multiple gradient encoded spin echo, and three-dimensional balanced steady-state free precession acquisitions. This method may provide a practical alternative to selective radiofrequency excitation at ultra-high-field, particularly for steady-state applications where repetition time (TR) must be minimized and when the amount of energy deposited in human tissues is prohibitive. There are several limitations to the approach including the spatial variation in resolution, high frequency aliasing artifacts, and spatial variation in echo times and contrast.

Multi-agent collaborative search: an agent-based memetic multi-objective optimization algorithm applied to space trajectory design

This article presents an algorithm for multi-objective optimization that blends together a number of heuristics. A population of agents combines heuristics that aim at exploring the search space both globally and in a neighbourhood of each agent. These heuristics are complemented with a combination of a local and global archive. The novel agent-based algorithm is tested at first on a set of standard problems and then on three specific problems in space trajectory design. Its performance is compared against a number of state-of-the-art multi-objective optimization algorithms that use the Pareto dominance as selection criterion: non-dominated sorting genetic algorithm (NSGA-II), Pareto archived evolution strategy (PAES), multiple objective particle swarm optimization (MOPSO), and multiple trajectory search (MTS). The results demonstrate that the agent-based search can identify parts of the Pareto set that the other algorithms were not able to capture. Furthermore, convergence is statistically better although the variance of the results is in some cases higher.

Design and validation of inverse optimisation software for the attitude control of microsatellites

The capabilities of microsatellite attitude control hardware have considerably evolved during the last two decades. However, three axis attitude control software is still predominantly based on the conservative use of standard flight proven PD type controllers, which are known to be limited in terms of rapidity for a prescribed level of energy consumption. Microsatellites are therefore typically not as agile as they could be. This conservatism is due to the complexity of implementing global numerical optimisation techniques to satellite attitude control. In this paper, we consider the model of a low earth orbiting microsatellite with a four wheel configuration, where the speed of one of the wheels is kept constant to provide a momentum bias and guarantee gyroscopic stiffness to disturbances. A geometric optimal control approach is presented, which circumvents the tedious tasks of numerically solving online the nonlinear optimisation problem. The approach is based on the design of suboptimal phase space trajectories. The phase space trajectory of a standard linear controller, typically a PD law with gyro-compensation, is used as a benchmark. The proposed inverse optimal control technique is then used to enforce higher convergence rate constraints than the benchmark law, without increasing the total energy consumption. The convergence rate of a Lyapunov function under the effect of the optimal controller outperforms the convergence rate of the same function under PD control and keeps increasing until a design settling time limit is reached. Guidelines are given for the tuning of the controller. The optimal attitude control algorithms are validated on a microsatellite software simulator in collaboration with the space company Surrey Satellite Technology Limited (SSTL). The software simulator incorporates a precise model of the effects of estimation errors, noise, external disturbances, sampling and actuator dynamics. The software is similar to the flight software of typical Surrey microsatellites. The proposed techniques are characterised by low implementation complexity because the difficulty is confined to the theoretical design stage. Settling time is significantly enhanced for the same level of energy consumed as the PD type law, which was used as a benchmark without loss of generality.

An Inflationary PSwarm Algorithm for Space Trajectory Design

A hybrid optimization algorithm, we call it Inflationary PSwarm Algorithm, is applied to solve the space trajectory design with gravity assists and multiple impulses. In this paper we will consider some simplified models proposed by the European Space Agency as test problems for global optimization. PSwarm is a pattern search method that incorporates a particle swarm scheme for dissemination of points in the feasible region in the (optional) search step. PSwarm is highly competitive with other global optimization methods based only on function values. An enhanced PSwarm with the capability of finding a global minimizer is proposed in this paper. The experiment results show the good performance obtained with our method and the improvement in terms of efficiency and computational cost.

Computing the Set of Epsilon-Efficient Solutions in Multiobjective Space Mission Design

In this work, we consider multiobjective space mission design problems. We will start from the need, from a practical point of view, to consider in addition to the (Pareto) optimal solutions also nearly optimal ones. In fact, extending the set of solutions for a given mission to those nearly optimal significantly increases the number of options for the decision maker and gives a measure of the size of the launch windows corresponding to each optimal solution, i.e., a measure of its robustness. Whereas the possible loss of such approximate solutions compared to optimal—and possibly even ‘better’—ones is dispensable. For this, we will examine several typical problems in space trajectory design—a biimpulsive transfer from the Earth to the asteroid Apophis and two low-thrust multigravity assist transfers—and demonstrate the possible benefit of the novel approach. Further, we will present a multiobjective evolutionary algorithm which is designed for this purpose.

New Lambert Algorithm Using the Hamilton-Jacobi-Bellman Equation

T HE two-point boundary-value problem (TPBVP) for Keplerian motion,which is known asLambert’s problem, is a fundamental problem in space trajectory design. Lambert’s problemmay be stated as follows: given initial and final position vectors, determine the initial velocity which will allow a transfer in a specified flight time. The classical approach to Lambert’s problem is based on geometry of the radius vectors, which directly stems from Lambert’s theorem [1]. After the early works of Gauss [2], a number of numerical algorithms were proposedbyLancaster et al. [3],Battin [4], andother researchers [5–7]. However, these solution procedures cannot be applied to the perturbed problem, because the geometry of the dynamics changes. On theother hand, general TPBVPs are solvedbymeans of numerical methods such as the shooting method and the finite difference method. Lizia et al. [8] and Armellin and Topputo [9] developed an integration scheme for the solution of the TPBVP and astrodynamics applications concerning the computation of trajectories in thevicinity of the libration points in the restricted threeand four-body problem. Guibout and Scheeres [10] solved the TPBVP using the theory of canonical transformation. These approaches [8–10] rely on a power series expansion along a nominal trajectory. Hamilton’s principle [11] is an alternative formulation of the differential equations of motion of spacecraft, which states that the trajectory between two specified states at two specified times is an extremum of the action integral. Motivated by this observation, this paper attempts to show that a solution toLambert’sproblemisdirectly obtained by minimizing the action integral. This problem can be viewedasanoptimalcontrolproblembyreplacingkineticenergywith a quadratic performance index of the control input, so that the initial velocity is found as the optimal control. Then, the solution is given by the Hamilton–Jacobi–Bellman (HJB) equation. The approximation methods to the solutions to the HJB equation were studied by many authors based on series expansion techniques [12–14]. In [14], a closed-formsolutionof theHJBequation isobtainedbyexpandingthe value function as a power series in terms of the state and the constant Lagrange multipliers. Although higher-order approximations are possible to obtain by series expansion solutions, their computations are time-consuming and there is no guarantee that the resulting solution improve the performance. Another approach is through successive approximation, where the HJB equation is reduced to a sequence of the first-order linear partial differential equations [15– 17]. Mizuno and Fujimoto [18] showed that the HJB equation is effectively solved by the Galerkin spectral method with Chebyshev polynomials based on successive approximation. In this paper, the TPBVP of the Hamiltonian system is treated as an optimal control problemwhere the Lagrangian function plays a role as a performance index. Similar toMizuno and Fujimoto, our approach is based on the expansionof thevalue function in theChebyshevserieswithunknown coefficients, considering the computational advantages of the use of Chebyshev polynomials. The differential expressions that arise in the HJB equation are also expanded in Chebyshev series with the unknown coefficients. As a consequence, the algorithm is much simpler than the procedure based on series expansion, and higherorder approximations are possible to obtain for more complicated nonlinear dynamics. Our algorithm can provide a solution to the TPVBP using the spectral information about the gravitation potential function and is also applicable to the TPVBP under a higher-order perturbed potential function without any modification. The paper is organized as follows. Section II is the problem statement. Section III.A reviews the Hamilton principle and the motivation of our method, Sec. III.B introduces the HJB equation in optimal control theory, and Sec. III.C formulates the solution procedure. Section IV presents simulation results for the two-body problem and the circular restricted three-body problem.

Analysis of Some Global Optimization Algorithms for Space Trajectory Design

Covers advancements in spacecraft and tactical and strategic missile systems, including subsystem design and application, mission design and analysis, materials and structures, developments in space sciences, space processing and manufacturing, space operations, and applications of space technologies to other fields.

A multi-objective approach to the design of low thrust space trajectories using optimal control

In this article, we introduce a novel three-step approach for solving optimal control problems in space mission design. We demonstrate its potential by the example task of sending a group of spacecraft to a specific Earth L_2 halo orbit. In each of the three steps we make use of recently developed optimization methods and the result of one step serves as input data for the subsequent one. Firstly, we perform a global and multi-objective optimization on a restricted class of control functions. The solutions of this problem are (Pareto-)optimal with respect to ΔV and flight time. Based on the solution set, a compromise trajectory can be chosen suited to the mission goals. In the second step, this selected trajectory serves as initial guess for a direct local optimization. We construct a trajectory using a more flexible control law and, hence, the obtained solutions are improved with respect to control effort. Finally, we consider the improved result as a reference trajectory for a formation flight task and compute trajectories for several spacecraft such that these arrive at the halo orbit in a prescribed relative configuration. The strong points of our three-step approach are that the challenging design of good initial guesses is handled numerically by the global optimization tool and afterwards, the last two steps only have to be performed for one reference trajectory.

A global optimization method for the design of space trajectories

The problem of optimally designing a trajectory for a space mission is considered in this paper. Actual mission design is a complex, multi-disciplinary and multi-objective activity with relevant economic implications. In this paper we will consider some simplified models proposed by the European Space Agency as test problems for global optimization (GTOP database). We show that many trajectory optimization problems can be quite efficiently solved by means of relatively simple global optimization techniques relying on standard methods for local optimization. We show in this paper that our approach has been able to find trajectories which in many cases outperform those already known. We also conjecture that this problem displays a "funnel structure" similar, in some sense, to that of molecular optimization problems.

A memetic multi-agent collaborative search for space trajectory optimisation

This paper presents the hybridisation of the fundamental heuristics underneath monotonic basin hopping within the general scheme of multi-agent collaborative search. The basic idea is that the local search performed by each individual agent in multi-agent collaborative search can be substituted with an iteration of basin hopping. Moreover, the local minima that are found during the search process are stored in an archive and at each iteration, the solution vector associated to each agent is extracted from the archive. The new hybrid algorithm is tested on some typical problems in space trajectory design and compared to monotonic basin hopping, a previous implementation of multi-agent collaborative search and to some standard evolutionary algorithms.

Application of high order expansions of two-point boundary value problems to astrodynamics

Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers.

A hybrid multiagent approach for global trajectory optimization

In this paper we consider a global optimization method for space trajectory design problems. The method, which actually aims at finding not only the global minimizer but a whole set of low-lying local minimizers (corresponding to a set of different design options), is based on a domain decomposition technique where each subdomain is evaluated through a procedure based on the evolution of a population of agents. The method is applied to two space trajectory design problems and compared with existing deterministic and stochastic global optimization methods.