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arxiv: 2605.20138 · v1 · pith:GKQ3VJ4Unew · submitted 2026-05-19 · 💻 cs.RO · cs.SY· eess.SY

Hamilton--Jacobi Reachability for Spacecraft Collision Avoidance

Larry Hui , Jordan Kam , William Su , Jianshu Zhou This is my paper

Pith reviewed 2026-05-20 04:33 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords spacecraft collision avoidanceHamilton-Jacobi reachabilityHill-Clohessy-Wiltshire dynamicsdifferential gamesbackward reachable setsorbital safetyRTN frameevasive maneuvers
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The pith

Backward reachable sets from Hamilton-Jacobi reachability identify relative states where satellite collisions can be provably avoided despite worst-case disturbances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies Hamilton-Jacobi reachability analysis to a two-satellite collision avoidance scenario in the same circular orbit. Relative motion is modeled using planar Hill-Clohessy-Wiltshire dynamics in the radial-tangential-normal frame. The problem is cast as a zero-sum differential game with one satellite controlled and the other as a bounded adversarial disturbance. Backward reachable sets are computed to mark unsafe configurations where collision cannot be avoided, while states outside these sets allow collision-free trajectories under any disturbance within bounds. This provides a foundation for hybrid control that triggers evasive maneuvers only when necessary, supporting scalable safety in orbital operations.

Core claim

The authors formulate the spacecraft interaction as a zero-sum differential game under Hill-Clohessy-Wiltshire dynamics and compute backward reachable sets that characterize relative states from which collision cannot be avoided under worst-case disturbances; states outside this set admit provably collision-free trajectories.

What carries the argument

Backward reachable sets computed via the Hamilton-Jacobi-Isaacs equation for the differential game, which delineate the boundary between avoidable and unavoidable collision regions in the relative state space.

If this is right

  • States outside the backward reachable set guarantee collision-free trajectories under any bounded disturbance from the other spacecraft.
  • The reachable sets integrate with supervisory hybrid control logic to determine when evasive maneuvers must be initiated.
  • Mathematically grounded safety guarantees enable scalability to multiple satellites.
  • The framework applies to minimum separation requirements consistent with FCC orbital standards.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar reachability methods could apply to avoiding orbital debris or managing larger satellite constellations.
  • Extensions might incorporate three-dimensional dynamics or time-varying orbits for broader applicability.
  • Real-time computation of these sets could support onboard decision-making in autonomous spacecraft.

Load-bearing premise

The relative motion between the two satellites is accurately modeled by planar Hill-Clohessy-Wiltshire dynamics assuming they are in the same circular orbit, and the second spacecraft behaves as a bounded disturbance with unknown intent.

What would settle it

A simulation or flight test showing a collision occurring from a relative state classified as safe (outside the backward reachable set) when the second spacecraft applies a disturbance within its modeled bounds would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.20138 by Jianshu Zhou, Jordan Kam, Larry Hui, William Su.

Figure 1
Figure 1. Figure 1: Relative-motion geometry in the RTN frame showing the target [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Escape maneuvers for the satellite fallback scenario. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 2D Capture Set for Recovery qeva to qret slice at 3 radial velocities. a finite number of grid cells, and each cell is assigned a numerical value of the level set function during the backward integration. We note that for the hybrid system automaton defined in Section III-C, the set of disturbance inputs is empty. To generate the reachable sets for a particular feedback control law u = K(X) as defined in S… view at source ↗
Figure 4
Figure 4. Figure 4: 3D Capture Set for Recovery qeva to qret slice at 3 radial velocities [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a Hamilton-Jacobi reachability framework for two-satellite collision avoidance in the same circular orbit. Relative motion is modeled via planar Hill-Clohessy-Wiltshire dynamics in the RTN frame. The interaction is cast as a zero-sum differential game with the controlled satellite as Player 1 and a bounded adversarial disturbance (unknown intent) as Player 2. Backward reachable sets are computed to identify states from which collision cannot be avoided under worst-case disturbances; states outside this set are claimed to admit provably collision-free trajectories. These sets are integrated with supervisory hybrid control logic to trigger evasive maneuvers, providing formal safety guarantees.

Significance. If the central claims hold, the work supplies a rigorous, game-theoretic method for certifying collision avoidance under adversarial disturbances in orbital scenarios, which is relevant for autonomous space operations and regulatory compliance with separation standards. A strength is the direct application of standard HJ-Isaacs theory to the HCW dynamics without introducing extraneous parameters beyond the disturbance bound; the safety interpretation follows by construction from the differential-game definition of the backward reachable set.

major comments (2)
  1. [§4] §4 (Hybrid Control Integration): The manuscript states that the reachable sets are 'integrated with supervisory hybrid control logic' to determine when evasive maneuvers must be initiated and that this enables 'mathematically grounded safety guarantees for scalability.' However, no explicit switching conditions, mode definitions, or invariance proof are provided showing that the closed-loop system remains outside the target set whenever the state is outside the BRS. This integration is load-bearing for the scalability and guarantee claims.
  2. [§3.2] §3.2 (Numerical Computation): The paper reports computed backward reachable sets but provides no error bounds, grid-resolution study, or verification against the analytic HCW flow that the numerical solution of the HJ-Isaacs PDE accurately approximates the true reachable set under the stated bounded disturbance. Without this, the practical interpretation of 'provably collision-free' trajectories rests on unquantified numerical error.
minor comments (3)
  1. [§2] The target set definition (minimum separation per FCC standards) is introduced in the abstract and §2 but never given an explicit mathematical expression (e.g., a ball in relative position coordinates). Adding this would clarify the unsafe set for readers.
  2. [§3] Notation for the value function and the Isaacs Hamiltonian is introduced without a dedicated table or consistent reference to prior HJ literature (e.g., Mitchell et al. or Tomlin et al.). A short notation summary would improve readability.
  3. [Figures 3-5] Figure captions for the reachable-set plots do not state the disturbance bound value or the time horizon used, making it difficult to reproduce or interpret the visualized sets.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We provide point-by-point responses to the major comments below and outline the revisions we plan to implement.

read point-by-point responses

  1. Referee: [§4] §4 (Hybrid Control Integration): The manuscript states that the reachable sets are 'integrated with supervisory hybrid control logic' to determine when evasive maneuvers must be initiated and that this enables 'mathematically grounded safety guarantees for scalability.' However, no explicit switching conditions, mode definitions, or invariance proof are provided showing that the closed-loop system remains outside the target set whenever the state is outside the BRS. This integration is load-bearing for the scalability and guarantee claims.

    Authors: We agree that the integration section would benefit from more explicit details to support the safety guarantee claims. In the revised manuscript, we will expand §4 to include: (1) formal definitions of the hybrid modes (e.g., nominal control and evasive maneuver mode), (2) the switching condition based on the boundary of the computed backward reachable set, and (3) a sketch of the invariance proof demonstrating that trajectories starting outside the BRS remain collision-free under the supervisory logic. This will make the formal guarantees more rigorous and address the scalability aspect. revision: yes

  2. Referee: [§3.2] §3.2 (Numerical Computation): The paper reports computed backward reachable sets but provides no error bounds, grid-resolution study, or verification against the analytic HCW flow that the numerical solution of the HJ-Isaacs PDE accurately approximates the true reachable set under the stated bounded disturbance. Without this, the practical interpretation of 'provably collision-free' trajectories rests on unquantified numerical error.

    Authors: We acknowledge the importance of validating the numerical accuracy of the HJ reachability computations. In the revision, we will add to §3.2 a grid convergence study showing how the reachable set boundaries stabilize with increasing grid resolution. We will also include a comparison of the numerical results with the analytic solution of the HCW equations in the absence of disturbances to quantify the discretization error. Error bounds will be discussed in the context of the Lipschitz continuity assumptions and the viscosity solution properties of the HJ-Isaacs equation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper applies the standard Hamilton-Jacobi reachability framework for zero-sum differential games to planar HCW dynamics with bounded adversarial disturbance. The backward reachable set is defined directly from the target unsafe set and the Isaacs PDE; its complement therefore admits collision-free trajectories by the external properties of the reachability operator, without any reduction to a fitted parameter, self-referential equation, or load-bearing self-citation. The FCC separation requirements and hybrid control logic are external inputs, and the central safety claim follows from the stated differential-game formulation rather than from any internal renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard orbital-mechanics assumptions and a bounded-disturbance model whose specific bounds are not detailed in the abstract.

free parameters (1)
  • disturbance bound
    The magnitude of the bounded adversarial disturbance is a modeling choice required to define the zero-sum game.
axioms (1)
  • domain assumption Planar Hill-Clohessy-Wiltshire equations accurately describe relative motion for two spacecraft in the same circular orbit.
    Invoked to define the state space and target unsafe set in the RTN frame.

pith-pipeline@v0.9.0 · 5706 in / 1214 out tokens · 33843 ms · 2026-05-20T04:33:38.705414+00:00 · methodology

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Reference graph

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