Enumeration of the Multiple-Outage Missed-Thrust Problem

Enumeration of the problem space for the multiple-outage missed-thrust problem is a significant factor for determining what types of analysis is feasible to conduct. An exhaustive examination of the multiple-outage problem is one method for conducting missed-thrust analysis. In this paper, we explore the combinatorics of the problem, providing methods of calculating and estimating the number of possible missed-thrust cases when discretizing a trajectory to consider the effects of multiple missed-thrust events. We examine the difficulties of enumerating all possible cases, and examine when it may be desirable to conduct an exhaustive enumeration rather than a stochastic Monte-Carlo-type analysis. For missions which cover large time scales or feature high resolutions for their analysis, the combinatorics of the problem quickly grow excessively large, but the exhaustive formulation can still provide useful insight for smaller problems or with the right assumptions. We also explore various sub-problems, and their enumeration, such as the partial-thrust problem and the consideration of only critical regions of the trajectory, their potential applicability to the field of missed-thrust analysis, and connections to mathematical topics.