Hamilton--Jacobi Reachability for Spacecraft Collision Avoidance

2026-05-19

AI Summary

This study addresses collision avoidance for co-orbital spacecraft pairs in the RTN plane under the Federal Communications Commission (FCC) minimum separation standard, focusing on worst-case threats arising from unknown but bounded disturbances. By modeling the relative motion as a zero-sum differential game within the Hamilton-Jacobi (HJ) reachability framework, the work identifies unsafe states leading to inevitable collisions via backward reachable sets and devises a hybrid supervisory control logic to trigger evasive maneuvers. It is the first to integrate HJ reachability analysis with FCC orbital safety criteria, yielding a mathematically guaranteed avoidance strategy. Within the planar Hill–Clohessy–Wiltshire dynamics, the approach explicitly delineates the boundary between safe and unsafe states, ensuring compliance with safety requirements despite disturbances and enabling scalable deployment.

Abstract

This article presents a Hamilton--Jacobi (HJ) reachability framework for a two--satellite collision avoidance problem operating in the same circular orbit, where relative motion is modeled in the radial--tangential--normal (RTN) frame using planar Hill--Clohessy--Wiltshire (HCW) dynamics. We define the target state space as unsafe relative configurations in the orbit plane corresponding to minimum separation requirements consistent with Federal Communications Commission (FCC) orbital standards. The interaction between spacecraft is formulated as a zero--sum differential game, where Player 1 is the controlled satellite and Player 2 is modeled as a bounded adversarial disturbance with unknown intent. We present the HJ formulation and compute backward reachable sets that characterize relative states from which collision cannot be avoided under worst-case disturbances, while states outside this set admit provably collision-free trajectories. These reachable sets are integrated with supervisory hybrid control logic to determine when evasive maneuvers must be initiated, enabling mathematically grounded safety guarantees for scalability.

Problem

• collision avoidance
• Hamilton-Jacobi reachability
• spacecraft
• differential game
• safety guarantees

Innovation

• Hamilton-Jacobi reachability
• collision avoidance
• differential game
• backward reachable set
• hybrid control