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Autonomous Constellation Fault Monitoring with Inter-satellite Links: A Rigidity-Based Approach

Keidai Iiyama, Daniel Neamati, Grace Gao

TL;DR

This work tackles autonomous fault detection for lunar satellite constellations in the absence of dense ground stations. It introduces a rigidity-based framework that uses two-way inter-satellite ranging and the geometric-centered Euclidean Distance Matrix ($GCEDM$) to detect faulty satellites without relying on precise ephemeris; the approach hinges on $2$-vertex redundantly rigid graph topology and analyzes the ranks of EDMs/GCEDMs to justify using the 4th and 5th singular values as fault indicators with a gamma-style test statistic. A clique-based online detector aggregates information across multiple subgraphs to identify and remove faulted satellites, with several hyperparameters guiding detection sensitivity and reliability. The authors validate the method on a Moon-centered Elliptical Lunar Frozen Orbit (ELFO) constellation and demonstrate robust fault detection performance across configurations, highlighting the framework’s potential for autonomous LunaNet integrity monitoring and future distributed implementations.

Abstract

To address the need for robust positioning, navigation, and timing services in lunar environments, this paper proposes a novel fault detection framework for satellite constellations using inter-satellite ranging (ISR). Traditionally, navigation satellites can depend on a robust network of ground-based stations for fault monitoring. However, due to cost constraints, a comprehensive ground segment on the lunar surface is impractical for lunar constellations. Our approach leverages vertex redundantly rigid graphs to detect faults without relying on precise ephemeris. We model satellite constellations as graphs where satellites are vertices and inter-satellite links are edges. We identify faults through the singular values of the geometric-centered Euclidean distance matrix (GCEDM) of 2-vertex redundantly rigid sub-graphs. The proposed method is validated through simulations of constellations around the Moon, demonstrating its effectiveness in various configurations. This research contributes to the reliable operation of satellite constellations for future lunar exploration missions.

Autonomous Constellation Fault Monitoring with Inter-satellite Links: A Rigidity-Based Approach

TL;DR

This work tackles autonomous fault detection for lunar satellite constellations in the absence of dense ground stations. It introduces a rigidity-based framework that uses two-way inter-satellite ranging and the geometric-centered Euclidean Distance Matrix () to detect faulty satellites without relying on precise ephemeris; the approach hinges on -vertex redundantly rigid graph topology and analyzes the ranks of EDMs/GCEDMs to justify using the 4th and 5th singular values as fault indicators with a gamma-style test statistic. A clique-based online detector aggregates information across multiple subgraphs to identify and remove faulted satellites, with several hyperparameters guiding detection sensitivity and reliability. The authors validate the method on a Moon-centered Elliptical Lunar Frozen Orbit (ELFO) constellation and demonstrate robust fault detection performance across configurations, highlighting the framework’s potential for autonomous LunaNet integrity monitoring and future distributed implementations.

Abstract

To address the need for robust positioning, navigation, and timing services in lunar environments, this paper proposes a novel fault detection framework for satellite constellations using inter-satellite ranging (ISR). Traditionally, navigation satellites can depend on a robust network of ground-based stations for fault monitoring. However, due to cost constraints, a comprehensive ground segment on the lunar surface is impractical for lunar constellations. Our approach leverages vertex redundantly rigid graphs to detect faults without relying on precise ephemeris. We model satellite constellations as graphs where satellites are vertices and inter-satellite links are edges. We identify faults through the singular values of the geometric-centered Euclidean distance matrix (GCEDM) of 2-vertex redundantly rigid sub-graphs. The proposed method is validated through simulations of constellations around the Moon, demonstrating its effectiveness in various configurations. This research contributes to the reliable operation of satellite constellations for future lunar exploration missions.
Paper Structure (27 sections, 7 theorems, 22 equations, 14 figures, 2 tables, 2 algorithms)

This paper contains 27 sections, 7 theorems, 22 equations, 14 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Required Graph Topology for Fault Detection
  3. Definitions
  4. Range Measurements and Satellite Faults
  5. Weighed Graph and Embeddability
  6. Rigid and vertex redundantly rigid graphs
  7. Embeddability of Graphs with Fault Satellites
  8. Sufficient condition for fault disprovability
  9. Necessary condition for fault disprovability
  10. Summary
  11. Properties of the Euclidean Distance Matrix
  12. Geometric Centered Euclidean Distance Matrix
  13. Rank of the Euclidean Distance Matrices with Fault Satellites
  14. Distribution of the Singular Values of the Geometric Centered EDM
  15. Redundantly Rigid Graph-Based Fault Detection
  16. ...and 12 more sections

Key Result

Proposition II.1

Suppose there is a single fault satellite on a graph $G' = \langle V, E, W \rangle$ that is k-vertex ($k \geq 2$) redundantly rigid in $\mathbb{R}^{d}$. If $G$ is $d$-embeddable, then $G$ contains no fault vertex (satellite) with probability 1.

Figures (14)

  • Figure 1: Flexible Graph
  • Figure 2: Rigid Graph
  • Figure 3: 2-vertex redundantly rigid graph
  • Figure 5: A visual description of the proof of Proposition \ref{['prop:fd_redudantly_rigd']} for $d=2$.
  • Figure 6: A visual description of the proof of Proposition \ref{['prop:fd_not_redudantly_rigd']} for $d=2$. We cannot detect faults by looking at the embeddability of graphs that are not 2-edge redundantly rigid.
  • ...and 9 more figures

Theorems & Definitions (16)

  • Proposition II.1
  • proof
  • Proposition II.2
  • proof
  • Proposition II.3
  • proof
  • Proposition II.4
  • Proposition III.1
  • proof
  • Proposition III.2
  • ...and 6 more