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Satellite Safe Margin: Fast solutions for Conjunction Analysis

Ricardo N. Ferreira, Marta Guimarães, Cláudia Soares

TL;DR

The paper addresses conjunction analysis under ellipsoidal uncertainty by introducing the margin, the smallest possible distance between two objects given their Gaussian position uncertainties. It provides two fast optimization-based solutions: a centralized Frank-Wolfe method and a distributed Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to compute the margin without sharing sensitive covariances. Across extensive experiments on real CDM data, the methods achieve high accuracy (FW up to 0.8 m, FISTA up to 0.2 m) and substantial speed advantages over a CVXPY ground truth, while highlighting that the classical Rimon-Boyd approach can suffer large errors due to non-normal matrices. The margin offers a physically interpretable, complementary tool to miss distance and probability of collision for operator decision-making in conjunction assessment, enabling scalable, privacy-preserving collaboration and robust risk evaluation.

Abstract

The amount of debris in orbit has increased significantly over the years. With the recent growth of interest in space exploration, conjunction assessment has become a central issue. One important metric to evaluate conjunction risk is the miss distance. However, this metric does not intrinsically take into account uncertainty distributions. Some work has been developed to consider the uncertainty associated with the position of the orbiting objects, in particular, to know if these uncertainty distributions overlap (e.g., ellipsoids when considering Gaussian distributions). With this work, we present fast solutions to not only check if the ellipsoids overlap but to compute the distance between them, which we call margin. We present two fast solution methods for two different paradigms: when the best-known data from both objects can be centralized (e.g., debris-satellite conjunctions) and when the most precise covariances cannot be shared (conjunctions of satellites owned by different operators). Our methods are both accurate and fast, being able to process 15,000 conjunctions per minute with the centralized solution and approximately 490 conjunctions per minute with the distributed solution.

Satellite Safe Margin: Fast solutions for Conjunction Analysis

TL;DR

The paper addresses conjunction analysis under ellipsoidal uncertainty by introducing the margin, the smallest possible distance between two objects given their Gaussian position uncertainties. It provides two fast optimization-based solutions: a centralized Frank-Wolfe method and a distributed Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to compute the margin without sharing sensitive covariances. Across extensive experiments on real CDM data, the methods achieve high accuracy (FW up to 0.8 m, FISTA up to 0.2 m) and substantial speed advantages over a CVXPY ground truth, while highlighting that the classical Rimon-Boyd approach can suffer large errors due to non-normal matrices. The margin offers a physically interpretable, complementary tool to miss distance and probability of collision for operator decision-making in conjunction assessment, enabling scalable, privacy-preserving collaboration and robust risk evaluation.

Abstract

The amount of debris in orbit has increased significantly over the years. With the recent growth of interest in space exploration, conjunction assessment has become a central issue. One important metric to evaluate conjunction risk is the miss distance. However, this metric does not intrinsically take into account uncertainty distributions. Some work has been developed to consider the uncertainty associated with the position of the orbiting objects, in particular, to know if these uncertainty distributions overlap (e.g., ellipsoids when considering Gaussian distributions). With this work, we present fast solutions to not only check if the ellipsoids overlap but to compute the distance between them, which we call margin. We present two fast solution methods for two different paradigms: when the best-known data from both objects can be centralized (e.g., debris-satellite conjunctions) and when the most precise covariances cannot be shared (conjunctions of satellites owned by different operators). Our methods are both accurate and fast, being able to process 15,000 conjunctions per minute with the centralized solution and approximately 490 conjunctions per minute with the distributed solution.

Paper Structure

This paper contains 20 sections, 43 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Setting the Benchmark: the Rimon-Boyd Approach
  4. Ellipse Overlap Test
  5. Margin Computation
  6. Optimal and Stable solution with data sharing: Iterative Frank-Wolfe algorithm
  7. Frank-Wolfe Algorithm
  8. Optimal and stable margin computation
  9. Iterative solution using Fast Iterative Shrinkage-Thresholding Algorithm
  10. Fast Iterative Shrinkage-Thresholding Algorithm
  11. Optimal and stable margin computation without shared covariances
  12. Projection onto the ellipse
  13. Numerical Experiments and Discussion
  14. Dataset
  15. Rimon-Boyd Benchmark
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Representation of the difference between the miss distance and the margin in two different conjunctions.
  • Figure 2: Two-dimensional representation of the $x^*$ and $y^*$ points for the margin computation between ellipses.
  • Figure 3: Cumulative Distribution Function of the difference between the miss distance and the margin computed with the Rimon and Boyd method.
  • Figure 4: Relative error of the margin value computed with the Rimon-Boyd Approach.
  • Figure 5: Relative error of the margin value computed with the Frank-Wolfe with Step-Size Line Search Approach.
  • ...and 2 more figures

Theorems & Definitions (2)

  • proof
  • proof