The 12th Global Trajectory Optimization Competition challenged teams to design trajectories for mining asteroids and transporting extracted resources back to the Earth. This paper outlines the methods and results of the runner-up team, BIT-CAS-DFH, highlighting an overall analysis of the approach as well as detailed descriptions of the methods used. The approach begins with building databases to reduce computational costs in trajectory design. Then, asteroid sequences are determined. A segmentation-based approach was adopted to efficiently handle the large dataset. Each sequence was divided into four time-based segments. Segments 1 and 4 were generated forward and backward, respectively, using a breadth-first beam search. Candidates for these segments were refined using genetic and greedy algorithms. Segments 2 and 3 were then generated and selected forward and backward based on the results of Segments 1 and 4. Following this, a matching process paired candidates from Segments 2 and 3. With the asteroid sequences established, low-thrust trajectories were optimized using indirect methods. A local optimization strategy was employed to maximize the collected mass by fine-tuning rendezvous timings. The final solution is presented, with comparative analyses against other teams' approaches.

Satellite encounters during close operations, such as rendezvous, formation, and cluster flights, are typical long-term encounters. The collision probability in such an encounter is a primary safety concern. In this study, a parametric method is proposed to compute the long-term collision probability for close satellite operations with initial state uncertainty. Random relative state errors resulting from system uncertainty lead to possible deviated trajectories with respect to the nominal one. To describe such a random event meaningfully, each deviated trajectory sample should be mapped to a unique and time-independent element in a random variable (RV) space. In this study, the RV space was identified as the transformed state space at a fixed initial time. The physical dimensions of both satellites were characterized by a combined hard-body sphere. Transforming the combined hard-body sphere into the RV space yielded a derived ellipsoid, which evolved over time and swept out a derived collision volume. The derived collision volume was solved using the reachable domain method. Finally, the collision probability was computed by integrating a probability density function over the derived collision volume. The results of the proposed method were compared with those of a nonparametric computation-intensive Monte Carlo method. The relative difference between the two results was found to be < 0.6%, verifying the accuracy of the proposed method.

Conventional reachable domain (RD) problem with an admissible velocity increment, $\mathrm{\Delta }𝐯$, in an isotropic distribution, was extended to the general case with $\mathrm{\Delta }𝐯$ in an anisotropic ellipsoidal distribution. Such an extension enables RD to describe the effect of initial velocity uncertainty because a Gaussian form of velocity uncertainty can be regarded as possible velocity deviations that are confined within an error ellipsoid. To specify RD in space, the boundary surface of RD, also known as the envelope, should be determined. In this study, the envelope is divided into two parts: inner and outer envelopes. Thus, the problem of solving the RD envelope is formulated into an optimization problem. The inner and outer reachable boundaries that are closest to and farthest away from the center of the Earth, respectively, were found in each direction. An optimal control policy is then formulated by using the necessary condition for an optimum; that is, the first-order derivative of the performance function with respect to the control variable becomes zero. Mathematical properties regarding the optimal control policy is discussed. Finally, an algorithm to solve the RD envelope is proposed. In general, the proposed algorithm does not require any iteration, and therefore benefits from quick computation. Numerical examples, including two coplanar cases and two 3D cases, are provided, which demonstrate that the proposed algorithm works efficiently.