Diffusion and Reactions in Fractals and Disordered Systems

There has been significant progress in the development of numerical methods for the determination of optimal trajectories for continuous dynamic systems, especially in the last 20 years In the 1980s, the principal contribution was new methods for discretizing the continuous system and converting the optimization problem into a nonlinear programming problem This has been a successful approach that has yielded optimal trajectories for very sophisticated problems In the last 15–20 years, researchers have applied a qualitatively different approach, using evolutionary algorithms or metaheuristics, to solve similar parameter optimization problems Evolutionary algorithms use the principle of “survival of the fittest” applied to a population of individuals representing candidate solutions for the optimal trajectories Metaheuristics optimize by iteratively acting to improve candidate solutions, often using stochastic methods In this paper, the advantages and disadvantages of these recently developed methods are described and an attempt is made to answer the question of what is now the best extant numerical solution method

A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems

an initial guess satisfying the equation of motion constraints and the terminal boundary conditions. A shape-based methodisderivedherethatiscapableofdeterminingsuchtrajectoriesforbothtime-freeandtime-fixedcases.When combined with a genetic algorithm, it is possible to find solutions by adjusting the free parameters of the problem within a few percent of optimal solutions found with a very exact optimizer. The method is intended for the rendezvous case but, with small modifications, it is able to solve for the less constrained cases of interception, orbit transfer, and escape.

Shape-Based Approach to Low-Thrust Rendezvous Trajectory Design

A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed optimization technique which uses a piecewise polynomial representation for the state and controls, and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

The rapid developments of artificial intelligence in the last decade are influencing aerospace engineering to a great extent and research in this context is proliferating. We share our observations on the recent developments in the area of spacecraft guidance dynamics and control, giving selected examples on success stories that have been motivated by mission designs. Our focus is on evolutionary optimisation, tree searches and machine learning, including deep learning and reinforcement learning as the key technologies and drivers for current and future research in the field. From a high-level perspective, we survey various scenarios for which these approaches have been successfully applied or are under strong scientific investigation. Whenever possible, we highlight the relations and synergies that can be obtained by combining different techniques and projects towards future domains for which newly emerging artificial intelligence techniques are expected to become game changers.

A Survey on Artificial Intelligence Trends in Spacecraft Guidance Dynamics and Control

Many space mission planning problems may be formulated as hybrid optimal control problems, that is, problems that include both real-valued variables and categorical variables. In orbital mechanics problems, the categorical variables will typically specify the sequence of events that qualitatively describe the trajectory or mission, and the real-valued variableswill represent the launchdate,flight times betweenplanets,magnitudes anddirections of rocket burns, flyby altitudes, etc. A current practice is to preprune the categorical state space to limit the number of possible missions to a number whose cost may reasonably be evaluated. Of course, this risks pruning away the optimal solution. Themethod to be developed here avoids the need for prepruning by incorporating a new solution approach. The new approach uses nested loops: an outer-loop problem solver that handles the finite dynamics and finds a solution sequence in terms of the categorical variables, and an inner-loop problem solver that finds the optimal trajectory for a given sequence A binary genetic algorithm is used to solve the outer-loop problem, and a cooperative algorithm based on particle swarm optimization and differential evolution is used to solve the inner-loop problem. Thehybrid optimal control solver is successfully demonstrated here by reproducing theGalileo andCassinimissions.

Automated Mission Planning via Evolutionary Algorithms

Preliminary design of low-thrust interplanetary missions is a highly complex process. The mission designer must choose discrete parameters such as the number of flybys, the bodies at which those flybys are performed, and in some cases the final destination. In addition, a time-history of control variables must be chosen that defines the trajectory. There are often many thousands, if not millions, of possible trajectories to be evaluated, which can be a very expensive process in terms of the number of human analyst hours required. An automated approach is therefore very desirable. This work presents such an approach by posing the mission design problem as a hybrid optimal control problem. The method is demonstrated on hypothetical missions to Mercury, the main asteroid belt, and Pluto.

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An Automated Solution of the Low-Thrust Interplanetary Trajectory Problem

Many optimization methods require accurate partial derivative information in order to ensure efficient, robust, and accurate convergence. This work outlines analytic methods for computing the problem Jacobian for two different bounded-impulse spacecraft trajectory models solved using two-sided shooting. The specific two-body Keplerian propagation method used by both of these models is described. Methods for incorporating realistic operational constraints and hardware models at the preliminary stage of a trajectory design effort are also demonstrated and the analytic methods derived are tested for accuracy using automatic differentiation. A companion paper will solve several relevant problems that show the utility of employing analytic derivatives, i.e. compared to using derivatives found using finite differences.

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Analytic Gradient Computation for Bounded-Impulse Trajectory Models Using Two-Sided Shooting.

Bounded-impulse trajectory models are an important component of many spacecraft trajectory preliminary design workflows. This paper includes discussions on practical implementation considerations f...

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Application and analysis of bounded-impulse trajectory models with analytic gradients

Trajectory optimization methods using monotonic basin hopping (MBH) have become well developed during the past decade [1, 2, 3, 4, 5, 6]. An essential component of MBH is a controlled random search through the multi-dimensional space of possible solutions. Historically, the randomness has been generated by drawing random variable (RV)s from a uniform probability distribution. Here, we investigate the generating the randomness by drawing the RVs from Cauchy and Pareto distributions, chosen because of their characteristic long tails. We demonstrate that using Cauchy distributions (as first suggested by J. Englander [3, 6]) significantly improves monotonic basin hopping (MBH) performance, and that Pareto distributions provide even greater improvements. Improved performance is defined in terms of efficiency and robustness. Efficiency is finding better solutions in less time. Robustness is efficiency that is undiminished by (a) the boundary conditions and internal constraints of the optimization problem being solved, and (b) by variations in the parameters of the probability distribution. Robustness is important for achieving performance improvements that are not problem specific. In this work we show that the performance improvements are the result of how these long-tailed distributions enable MBH to search the solution space faster and more thoroughly. In developing this explanation, we use the concepts of sub-diffusive, normally-diffusive, and super-diffusive random walks (RWs) originally developed in the field of statistical physics.

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Jacob A. Englander

Papers

Automated Mission Planning via Evolutionary Algorithms

An Automated Solution of the Low-Thrust Interplanetary Trajectory Problem

Analytic Gradient Computation for Bounded-Impulse Trajectory Models Using Two-Sided Shooting.

Application and analysis of bounded-impulse trajectory models with analytic gradients

Tuning Monotonic Basin Hopping: Improving the Efficiency of Stochastic Search as Applied to Low-Thrust Trajectory Optimization