  To address the need for robust positioning, navigation, and timing services
in lunar and Martian environments, this paper proposes a novel fault detection
framework for satellite constellations using inter-satellite ranging (ISR).
Traditional fault monitoring methods rely on intense monitoring from
ground-based stations, which are impractical for lunar and Martian missions due
to cost constraints. Our approach leverages graph-rigidity theory to detect
faults without relying on precise ephemeris. We model satellite constellations
as graphs where satellites are vertices and inter-satellite links are edges. By
analyzing the Euclidean Distance Matrix (EDM) derived from ISR measurements, we
identify faults through the singular values of the geometric-centered EDM
(GCEDM). A neural network predictor is employed to handle the diverse geometry
of the graph, enhancing fault detection robustness. The proposed method is
validated through simulations of constellations around Mars and the Moon,
demonstrating its effectiveness in various configurations. This research
contributes to the reliable operation of satellite constellations for future
lunar and Martian exploration missions.
