Optimizing multi-rendezvous spacecraft trajectories: $\Delta V$ matrices and sequence selection

Multi-rendezvous spacecraft trajectory optimization problems are notoriously difficult to solve. For this reason, the design space is usually pruned by using heuristics and past experience. As an alternative, the current research explores some properties of $\Delta V$ matrices which provide the minimum $\Delta V$ values for a transfer between two celestial bodies for various times of departure and transfer duration values. These can assist in solving multi-rendezvous problems in an automated way. The paper focuses on the problem of, given a set of candidate objects, how to find the sequence of $N$ objects to rendezvous with that minimizes the total $\Delta V$ required. Transfers are considered as single algebraic objects corresponding to $\Delta V$ matrices, which allow intuitive concatenation via a generalized summation. Waiting times, both due to mission requirements and prospects for cheaper and faster future transfers, are also incorporated in the $\Delta V$ matrices. A transcription of the problem as a shortest path search on a graph can utilize a range of available efficient shortest path solvers. Given an efficient $\Delta V$ matrix estimator, the new paradigm proposed here is believed to offer an alternative to the pruning techniques currently used.