GTOC-11: Results from the University of Alabama

Highlights

• GTOC11 results from the Astrodynamics and Space Research Laboratory (ASRL) team.

• Selection of asteroid population subsets based on mean deliverable mass to stations.

• Deterministic beam search algorithm with full-fidelity optimal control solutions.

• A bin packing algorithm for minimizing variance in final Dyson station mass.

Abstract

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.

Introduction

The 11th Global Trajectory Optimization Competition (GTOC-11) tasked 91 teams from around the world to construct a Dyson Ring using materials acquired from the asteroid belt [1], [2]. These asteroids would be used as building materials for a circular ring of Dyson stations, which could be used to generate power from the Sun. The problem required competitors to optimize 10 impulsive motherships to fly by as many asteroids as possible, from more than 83000 candidates. After a mothership encountered each asteroid, a low-thrust vehicle was attached to the asteroid so that it could be transferred to 1 of 12 potential Dyson Ring building stations. For reference, the objective function of the competition is given below in Eq. (1), with $B$ being the time factor bonus, dependent on the time of submission for a given solution.

$$J=\frac{B\ast \left(10^{-10}{M}_{\min }\right)}{a_{\mathrm{Dyson}}^2\sum _{k=1}^{10}{(1+\Delta V_k^{\mathrm{Total}}/50)}^2}$$$$B=2-\sqrt{1-{\left(1-\frac{t_{\mathrm{submission}}-t_{\mathrm{start}}}{t_{\mathrm{end}}-t_{\mathrm{start}}}\right)}^3}$$

The quantities of interest include $M_{\min }$, which is the final mass of the smallest of the 12 Dyson stations, and $a_{\mathrm{Dyson}}$, which is the radius of the Dyson Ring in AUs. Additionally, minimizing the amount of fuel used by each mothership, $\Delta V_k^{\mathrm{total}}$, was required. This set of equations provided a satisfyingly complex problem space, with many key decisions to make in terms of target asteroids, Dyson Ring radius, number of flybys made, Dyson station ordering, and where to devote effort throughout different portions of the competition to produce higher objective values.

The problem required that teams determine a set of orbital parameters defining the Dyson Ring where the stations would be placed. Rings placed closer to the Sun rewarded teams with a quadratically increased score, acting as a proxy for the increased power generation for such a ring. Additionally, the other orbital elements of the station had to be chosen, such as inclination and right ascension of the ascending node (RAAN). The ring was defined to be circular in the problem statement and therefore possesses an eccentricity of 0. The Dyson Ring must have 12 stations equally spaced around the orbit, and the stations were to be constructed one-by-one with no simultaneous construction allowed. The restrictions placed by the requirement to build the stations in order significantly increased the difficulty of the problem. Many asteroids were limited to which station they could approach due to time constraints on the building of the stations, such that a new station could only begin construction 90 days after the completion of the previous station. Similar constraints were placed on the activation of the asteroids, requiring that the asteroid transfers could only begin 30 days after the mothership’s rendezvous with a candidate asteroid. Lastly, the objective function’s reliance on minimum station mass required that effort be placed into designing the mothership tours and subsequent asteroid transfers to equally distribute asteroid mass between stations.

Beginning with a macroscopic level search, key parameters such as Dyson ring construction radius, inclination, and station build order were determined to reduce the search space of the problem. By computing average values over subsets of candidate populations, regions of space containing clusters of high-mass, low eccentricity and inclination asteroids were found. In order to explore this solution space, a beam search method was implemented. Beam searches have been used numerous times by competitors in previous GTOC competitions [3], [4] with great success. Using a “flexing” beam search algorithm developed during the competition, these candidate asteroid populations could be deterministically searched for low $\Delta V$ asteroid tours for each of the 10 motherships. The optimizer for the beam search utilizes the Astrodynamics Software and Science Enabling Toolkit (ASSET), which uses the Parallel Sparse Interior Point Optimizer (PSIOPT), both of which were developed at the University of Alabama in the Astrodynamics and Space Research Laboratory (ASRL) [5]. PSIOPT can be used to solve smooth non-linear programs in parallel which allowed for a broad search approach during the competition. Using ASSET, the flexing beam search utilized a full fidelity optimization step over the entire asteroid tour to ensure that the complete leg was optimized at each step, as opposed to locally optimizing only the intermediate trajectory between each candidate asteroid. The addition of asteroids to the tour leg resulted in the rendezvous with earlier asteroids to be shifted in location, causing the full asteroid tour to appear to “flex” in shape as the trajectory was developed. This provided a more accurate representation for the $\Delta V$ required for the asteroid tour legs, which would then be used later in a bin-packing algorithm to evenly distribute the asteroid mass between the Dyson stations.

With the full asteroid tour legs, the asteroids were then transferred to the Dyson stations using a low-thrust propulsion method, which consumed a fixed rate of the asteroids mass for each unit of time the engine was active. Therefore, to reduce the amount of mass consumed (thereby increasing the total mass delivered to the stations), a time minimizing optimization method was used to deliver the asteroids as quickly as possible. However, as the asteroid transfer time is dependent on the time of activation of the asteroid transfer device (ATD) a range of departure times for each asteroid were computed to identify the mass maximizing deployment time and optimal Dyson station for delivery. Using this range of ATD activation times and subsequent station deliveries, a robust bin-packing algorithm was constructed to minimize final station mass variation.

The final bin-packing algorithm utilized hundreds of completed asteroid tours and transfers to form combinations of 10 legs which could be scored to determine the full score for a particular combination. These legs were then substituted with other complete tours to determine which tours could be replaced in order to increase the final objective value. This process was repeated millions of times to evenly distribute asteroid mass over the 12 Dyson stations.

The organization of the remainder of the paper is as follows. First, an overview of the tools utilized throughout the competition is given, along with the initial approach for pre-analysis and resulting assumptions for forming the optimization problem. These assumptions and pre-analysis metrics are then used to formulate the asteroid mothership tours, including defining the Lambert solver time of flight, maximum allowable $\Delta V$ per mothership, and number of asteroids to include in each tour leg using the “flexing” beam search methodology. Following the construction of the asteroid tour legs, the approach to transferring the asteroids from the mothership tours down to the assumed Dyson station locations is given. The low-thrust optimization approach is explained, along with the mapping of deliverable asteroid mass to the Dyson station stations. Additionally, the bin-packing algorithm for maximization of the minimum asteroid mass is shown. Lastly, the best scoring submission by the team is presented, along with discussion on the maximum estimated score and the final achieved score during the competition.