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In this work, Team ASRL’s solution approach for the 11th Global Trajectory Optimization Competition (GTOC-11) is described. This problem tasked the competing teams with constructing a futuristic Dyson Ring utilizing materials acquired from the asteroid belt. In total, 10 motherships would depart from Earth in the year 2121 and visit as many asteroids as possible. After visiting each asteroid, a low-thrust propulsion module would transfer the material down to the desired final Dyson stations. The final approach utilized a deterministic tree search that involved alternating between fixed time of flight Lambert searches and solutions to the full-fidelity optimal control problem. Once a single tour had been constructed, transfer trajectories were computed for each asteroid to as many of the building stations as possible. After computing a pool of thousands of these completed legs, a bin packing algorithm was used to determine the highest scoring combination of 10 solutions. This search process was implemented in Python using the soon-to-be-released trajectory optimization tool, ASSET. Ultimately, the team finished 5th with a score of 5525.38.

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View Video Presentation: https://doi.org/10.2514/6.2022-1131.vid The Astrodynamics Software and Science Enabling Toolkit (ASSET) is a newly developed spacecraft trajectory optimizing toolkit for deep-space, lunar, and icy worlds exploration. ASSET brings together many practical tools for trajectory optimization into one comprehensive package. To facilitate this goal, ASSET features a modular design philosophy with all objects being implemented in an abstract, extensible form. This technique allows for a suite of new mission architectures to be explored, without having to develop a mission-specific tool. ASSET utilizes a heavily standardized, object-oriented code structure to enable faster mission development and ease the overhead of complex coding implementations. As such, ASSET features a front-end Python interface for quick and painless model implementations, as well as a C++ back-end for more computationally intensive features. ASSET is applicable to any trajectory optimization problem, whether the targets are single bodies, or more advanced concepts such as simultaneous multi-spacecraft, multi-target (MSMT) analysis. This software package is aimed to increase accessibility to astrodynamics concepts and tools, allowing for increased engagement from the scientific community on a standardized, robust toolkit.

The removal of 123 pieces of debris from the Sun-synchronous LEO environment is accomplished by a 10-spacecraft campaign wherein the spacecraft, flying in succession over an 8-yr period, rendezvous with a series of the debris objects, delivering a de-orbit package at each one before moving on to the next object by means of impulsive manoeuvres. This was the GTOC9 problem, as posed by the European Space Agency. The methods used by the Jet Propulsion Laboratory team are described, along with the winning solution found by the
team. Methods include branch-and-bound searches that exploit the natural nodal drift to compute long chains
of rendezvous with debris objects, beam searches for synthesising campaigns, ant colony optimisation, and a
genetic algorithm. Databases of transfers between all bodies on a fine time grid are made, containing an easy-to-compute yet accurate estimate of the transfer ∆V . Lastly, a final non-linear programming optimisation is performed to ensure the trajectories meet all the constraints and are locally optimal in initial mass.

The orbital boundary value problem, also known as Lambert Problem, is
revisited. Building upon Lancaster and Blanchard approach, new relations are
revealed and a new variable representing all problem classes, under
L-similarity, is used to express the time of flight equation. In the new
variable, the time of flight curves have two oblique asymptotes and they mostly
appear to be conveniently approximated by piecewise continuous lines. We use
and invert such a simple approximation to provide an efficient initial guess to
an Householder iterative method that is then able to converge, for the single
revoltuion case, in only two iterations. The resulting algorithm is compared to
Gooding's procedure revealing to be numerically as accurate, while having a
smaller computational complexity.

Global optimization algorithms and space pruning methods represent a recent new paradigm for spacecraft trajectory design. They promise an automated and unbiased search of different trajectory options, freeing the final user from the need for caring about implementation details. In this chapter we provide a unified framework for the definition of trajectory problems as pure mathematical optimization problems highlighting their common nature. We then present the detailed definition of two popular typologies, the Multiple Gravity Assist (MGA) and the Multiple Gravity Assist with single Deep Space Manouver (MGA-1DSM). Later we describe in detail the instantiation of four particular problems proposing them as a test set to benchmark the performances of different algorithms and pruning solutions. We take inspiration from real interplanetary trajectories such as Cassini, Rosetta, and the proposed TandEM mission, considering a large search space in terms of possible launch windows and transfer times, but also from rather academic cases such as that of the First Global Trajectory Optimisation Competition (GTOC). We test four popular heuristic paradigms on these problems (differential evolution, particle swarm optimization, simulated annealing with adaptive neighborhood, and genetic algorithm) and note their poor performances both in terms of reliability and solution quality, arguing for the need to use more sophisticated approaches, for example, pruning methods, to allow finding better trajectories. We then introduce the cluster pruning method for the MGA-1DSM problem and we apply it, in combination with the simulated annealing with adaptive neighborhood algorithm, to the TandEM test problem finding a large number of good solutions and a new putative global optima.

We discuss the relationships between three approaches to greedy heuristic search: best-first, hill-climbing, and beam search. We consider the design decisions within each family and point out their oft-overlooked similarities. We consider the following best-first searches: weighted A*, greedy search, A ∗ ǫ, window A * and multi-state commitment k-weighted A*. For hill climbing algorithms, we consider enforced hill climbing and LSS-LRTA*. We also consider a variety of beam searches, including BULB and beam-stack search. We show how to best configure beam search in order to maximize robustness. An empirical analysis on six standard benchmarks reveals that beam search and best-first search have remarkably similar performance, and outperform hill-climbing approaches in terms of both time to solution and solution quality. Of these, beam search is preferable for very large problems and best first search is better on problems where the goal cannot be reached from all states.

We discuss the relationships between three approaches to greedy heuristic search: best-first, hill-climbing, and beam search. We consider the design decisions within each family and point out their oft-overlooked similarities. We consider the following best-first searches: weighted A*, greedy search, ASeps, window A* and multi-state commitment k-weighted A*. For hill climbing algorithms, we consider enforced hill climbing and LSS-LRTA*. We also consider a variety of beam searches, including BULB and beam-stack search. We show how to best configure beam search in order to maximize robustness. An empirical analysis on six standard benchmarks reveals that beam search and best-first search have remarkably similar performance, and outperform hill-climbing approaches in terms of both time to solution and solution quality. Of these, beam search is preferable for very large problems and best first search is better on problems where the goal cannot be reached from all states.

The National University of Defense Technology and the Xi’an Satellite Control Center organized the 11th edition of the Global Trajectory Optimization Competition (GTOC11) in 2021. The GTOC11 problem was created as a link between the ninth and tenth editions of the Global Trajectory Optimization Competition to bridge the gap between the planetary and galactic civilizations by introducing an intermediate stellar civilization scenario. The problem involves the construction of a Dyson sphere, a theoretical mega-structure that encircles a star with platforms orbiting in a tight formation to capture the maximum energy from it. Challenges in astrodynamics including the construction-orbit selection, combinatorial flyby of multiple asteroids, and mass distribution among stations were considered in the Dyson sphere design. A total of 94 teams registered for the competition, of which 25 teams provided solutions and passed automatic verification on the website. In this article, we describe the selection of the problem and its design process. In addition, an overview of the entire competition and an analysis of its results are presented.

Modified equinoctial elements are introduced which are suitable for perturbation analysis of all kinds of orbit. Equations of motion in Lagrangian and Gaussian forms are derived. Identities connecting the partial derivatives of the disturbing function with respect to equinoctial elements are established. Numerical comparisons of the evolution of a perturbed, highly eccentric, elliptic orbit analysed in equinoctial elements and by Cowell's method show satisfactory agreement.

We consider the NP-complete problem of bin packing. Given a set of numbers, and a set of bins of fixed capacity, find the minimum number of bins needed to contain all the numbers, such that the sum of the numbers assigned to each bin does not exceed the bin capacity. We present a new algorithm for optimal bin packing. Rather than considering the different bins that each number can be placed into, we consider the dif- ferent ways in which each bin can be packed. Our algorithm appears to be asymptotically faster than the best existing op- timal algorithm, and runs more that a thousand times faster on problems with 60 numbers.

GTOC-11-dyson sphere

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Forum, San Diego, CA, pp. 2022-1131, http://dx.doi.org/10.2514/6.2022-1131,
URL https://arc.aiaa.org/doi/abs/10.2514/6.2022-1131.