Stereographic projection and static digraph for information routing in satellite constellations

This study presents new approaches that can be applied to large constellations for (a) the rapid determination of routing nodes and (b) identification of optimal routing paths in relation to specific objectives. Problem (a) is solved through the stereographic projection of the satellites orbits onto the equatorial plane, which leads to analytical formulas expressing the intervals of mutual visibility between the satellites. With regard to problem (b), a new approach is proposed that models information routing as a Markov decision process. States of the process correspond to inter-satellite links and the two possible actions for each state are represented by either (i) information sharing or (ii) avoiding information sharing. An appropriate reward is assigned to each action/state combination and the actions corresponding to the maximum reward (i.e., the optimal policy) are determined through a value iteration algorithm. Then, a Markov chain is generated, where the connecting arcs represent possible transitions between two states. This chain — also termed static digraph hence forward — considers only feasible one-way transitions on the basis of sequential mutual connectivity events, unlike typical approaches in the literature that introduce time-varying graphs, which instead consider each satellite as a node. In contrast, the static digraph incorporates all the connectivity events within a specified horizon. Using the static digraph, an effective routing algorithm is proposed, to identify the path associated with maximum reward, able to convey information between collection and terminal points. Different rewards are evaluated, in relation to distinct operational constraints or requirements, such as the need of minimizing the transfer time or limiting the hop-count. Simulations prove that the methodology at hand is effective for computing information sharing in large constellations.