Low-Thrust Orbital Differential Games with Speed Constraint Enforcement Using Cost Weighting

Vitaly Shaferman; Yahli Drucker

arxiv: 2606.17594 · v2 · pith:OBFAM4VVnew · submitted 2026-06-16 · 📡 eess.SY · cs.SY· math.OC

Low-Thrust Orbital Differential Games with Speed Constraint Enforcement Using Cost Weighting

Yahli Drucker , Vitaly Shaferman This is my paper

Pith reviewed 2026-06-26 23:14 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords differential gamespursuit-evasionspacecraft guidancelow-thrustlinear-quadratic gameterminal speed constraintoptimal controlsaddle-point solution

0 comments

The pith

Weighting parameters in a linear-quadratic differential game enforce any terminal relative speed while yielding closed-loop minimum-fuel guidance laws for spacecraft pursuit-evasion.

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

This paper models the terminal phase of low-thrust spacecraft pursuit-evasion near circular orbit as a linear-quadratic zero-sum differential game with soft terminal position and velocity constraints plus running control costs. It derives analytical closed-loop optimal guidance laws for each player that form a saddle-point solution and minimize fuel use. The central result is that any desired terminal relative speed is achieved simply by selecting the cost-function weighting parameters on terminal velocity. Conjugate-point analysis verifies optimality of the solution even when the velocity weighting matrix is negative definite, a case that appears at high terminal speeds.

Core claim

An analytical, closed-loop, minimum-fuel-consumption optimal guidance law is derived for each player, forming a saddle-point solution. It is proven that any terminal speed can be achieved by properly choosing the weighting parameters of the cost function. To verify the optimality of the solution, a conjugate point analysis is performed when the cost function velocity weighting matrix is either positive or negative definite. The negative-definite case arises at high terminal speeds and is seldom seen in the literature. The performance of the derived guidance law is evaluated in simulations for different target maneuvers and compared to a state-of-the-art optimal-control-based guidance law.

What carries the argument

Linear-quadratic zero-sum differential game with soft terminal position-velocity constraints and running control-effort costs, where terminal-velocity weighting parameters enforce the desired terminal speed without hard constraints.

If this is right

Each player obtains an explicit state-feedback guidance law that satisfies the saddle-point property.
Any prescribed terminal relative speed is attained by suitable choice of the terminal-velocity weighting matrix entries.
Conjugate-point analysis establishes that the solution remains optimal when the velocity weighting matrix is positive definite or negative definite.
Simulations confirm that the guidance laws meet the terminal constraints and consume less fuel than a comparable optimal-control law when the target maneuvers optimally.

Where Pith is reading between the lines

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

The same weighting technique could be tested on pursuit-evasion problems whose linearization is valid over longer time intervals than a single terminal phase.
Tuning the weights offers a direct way to trade terminal-speed requirements against total fuel without reformulating the game.
The approach may extend to three-dimensional or non-circular reference orbits if the underlying linear dynamics remain a good local approximation.

Load-bearing premise

The terminal-phase relative motion of two spacecraft near a circular orbit can be accurately captured by a linear-quadratic zero-sum differential game model with soft terminal constraints, allowing the cost-weighting parameters to enforce arbitrary terminal speeds without violating optimality.

What would settle it

Closed-loop numerical integration of the derived guidance laws in which the realized terminal relative speed deviates from the value predicted by the weighting-parameter formula, or in which total fuel consumption exceeds that of a direct numerical optimal-control solver when the evader follows its derived strategy.

read the original abstract

This paper considers the problem of a low-thrust spacecraft pursuit-evasion differential game with an arbitrary terminal relative speed constraint. It addresses the terminal phase of the engagement for two relatively close spacecraft near a circular orbit. The problem is formulated as a linear-quadratic zero-sum differential game, with soft constraints on the terminal relative position and velocity, and running costs on the players' control efforts. An analytical, closed-loop, minimum-fuel-consumption optimal guidance law is derived for each player, forming a saddle-point solution. It is proven that any terminal speed can be achieved by properly choosing the weighting parameters of the cost function. To verify the optimality of the solution, a conjugate point analysis is performed when the cost function velocity weighting matrix is either positive or negative definite. The negative-definite case arises at high terminal speeds and is seldom seen in the literature. The performance of the derived guidance law is evaluated in simulations for different target maneuvers and compared to a state-of-the-art optimal-control-based guidance law. The simulations show that the derived guidance law satisfies the constraints and offers a substantial advantage over the optimal-control-based guidance law when the target is optimally evading.

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.

Desk Editor's Note private letter to a colleague

The paper gives a closed-form LQ differential-game guidance law for low-thrust orbital pursuit-evasion that tunes arbitrary terminal relative speed through cost weighting and checks conjugate points for the negative-definite velocity case, but the linearization may limit applicability at high speeds.

read the letter

The main point is an analytical closed-loop guidance law from an LQ differential game for low-thrust spacecraft pursuit-evasion near circular orbits, where terminal relative speed is enforced by weighting the velocity term in the cost function, with conjugate-point verification for the negative-definite case.

What is new is the specific use of cost weighting to achieve arbitrary terminal speeds in this orbital setting, plus the analysis for negative weights that rarely appears in the literature. The simulations evaluate the law against various target maneuvers and show it meets the constraints while outperforming a state-of-the-art optimal-control guidance law when the target evades.

The derivation follows standard LQ game theory on linearized relative dynamics with soft terminal constraints, so the saddle-point solution is expected once the Riccati equations are solved. The claim that any terminal speed can be reached by tuning weights is the central result, backed by the conjugate-point work.

A potential issue is whether the linear model remains valid. As noted in the stress-test, high terminal speeds might drive the relative states outside the regime where the linearization is accurate, which could mean the derived law and the speed proof don't fully apply to the underlying nonlinear orbital mechanics. The paper would benefit from addressing this, perhaps with nonlinear simulations or limits on the weights.

This paper is aimed at researchers in orbital mechanics and differential games for space applications. It has a clear method, analysis, and numerical results, so it deserves peer review to check the details and the modeling assumptions.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates the terminal phase of a low-thrust spacecraft pursuit-evasion game near a circular orbit as a linear-quadratic zero-sum differential game with soft terminal constraints on relative position and velocity. It derives analytical closed-loop minimum-fuel optimal guidance laws for each player that constitute a saddle-point solution, proves that arbitrary terminal relative speeds are achievable by tuning the cost-function weighting parameters (including the negative-definite velocity-weighting case), performs conjugate-point analysis to confirm optimality in both positive- and negative-definite regimes, and evaluates the laws via simulation against a state-of-the-art optimal-control guidance law, showing constraint satisfaction and fuel advantages when the target evades optimally.

Significance. If the derivations hold, the work supplies an explicit parameter-tuning procedure to enforce arbitrary terminal speeds inside an LQ game framework, together with conjugate-point verification that covers the seldom-treated negative-definite velocity weighting. The analytical closed-loop laws and the explicit proof of speed achievability constitute a concrete, falsifiable contribution to orbital differential-game guidance; the simulation comparisons further supply reproducible performance data.

major comments (2)

[Formulation and proof sections (abstract claim and § on dynamics)] The central claim that any terminal speed is achievable by cost weighting rests on the Clohessy-Wiltshire linearization remaining valid over the entire trajectory. For the high-speed (negative-definite) cases the paper explicitly contemplates, relative velocities can grow rapidly enough to exit the small-perturbation regime; no section supplies a quantitative bound on admissible terminal speed or a direct comparison of the LQ law against the underlying nonlinear orbital dynamics.
[Conjugate-point analysis section] The conjugate-point analysis is presented for both positive- and negative-definite velocity weighting matrices, yet the manuscript does not state the explicit Riccati or Hamiltonian matrix whose eigenvalues are checked, nor the numerical procedure used to locate conjugate points when the weighting matrix is negative definite.

minor comments (2)

[Notation and cost functional] Notation for the terminal weighting matrices (position and velocity) should be introduced once and used consistently; several symbols appear to be redefined between the cost functional and the closed-loop law.
[Simulation results] The simulation section would benefit from an explicit statement of the integration method, time step, and initial-condition sampling used to generate the reported fuel-consumption and miss-distance statistics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [Formulation and proof sections (abstract claim and § on dynamics)] The central claim that any terminal speed is achievable by cost weighting rests on the Clohessy-Wiltshire linearization remaining valid over the entire trajectory. For the high-speed (negative-definite) cases the paper explicitly contemplates, relative velocities can grow rapidly enough to exit the small-perturbation regime; no section supplies a quantitative bound on admissible terminal speed or a direct comparison of the LQ law against the underlying nonlinear orbital dynamics.

Authors: We acknowledge that the validity of the Clohessy-Wiltshire linearization for large terminal speeds is an important consideration not explicitly bounded in the current manuscript. The work is formulated and solved entirely within the linear framework, which is standard for close-proximity terminal-phase guidance, and the reported simulations remain inside the regime where the linearization is reasonable. To strengthen the presentation, the revised manuscript will add a dedicated paragraph (or short subsection) that supplies a quantitative estimate of the terminal-speed range for which the linear model remains admissible, derived from the underlying relative-distance and velocity assumptions of the Clohessy-Wiltshire equations. revision: yes
Referee: [Conjugate-point analysis section] The conjugate-point analysis is presented for both positive- and negative-definite velocity weighting matrices, yet the manuscript does not state the explicit Riccati or Hamiltonian matrix whose eigenvalues are checked, nor the numerical procedure used to locate conjugate points when the weighting matrix is negative definite.

Authors: We agree that the conjugate-point section would benefit from greater explicitness. The analysis follows the standard Hamiltonian-matrix test for the linear-quadratic game, but the manuscript does not display the matrix or the numerical detection procedure. In the revision we will insert the explicit 12-by-12 Hamiltonian matrix (constructed from the system dynamics and the indefinite cost weighting) together with a concise description of the numerical procedure used to monitor its eigenvalues and locate conjugate points, including the handling of the negative-definite velocity-weighting case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from standard LQ game theory

full rationale

The paper starts from the standard linear-quadratic zero-sum differential game formulation on Clohessy-Wiltshire relative dynamics with soft terminal penalties and running control costs. The closed-loop saddle-point guidance laws are obtained by solving the associated Riccati differential equations, and the claim that arbitrary terminal speeds are achievable follows directly from varying the definiteness and magnitude of the terminal velocity weighting matrix together with a conjugate-point check. None of these steps reduce by construction to a fitted parameter or to a self-citation whose content is the target result itself; the analysis is mathematically self-contained once the LQ setup is accepted.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the linear-quadratic approximation being valid for the terminal phase and on the existence of weighting parameters that achieve any speed while preserving the saddle-point property; these are domain assumptions rather than derived quantities.

free parameters (1)

terminal velocity weighting matrix
Tuned to enforce a chosen terminal relative speed; the abstract states that any speed is achievable by proper choice of these parameters.

axioms (2)

domain assumption Relative dynamics near a circular orbit admit a linear time-invariant approximation suitable for LQ differential game formulation
Invoked to justify the linear-quadratic zero-sum game model with soft terminal constraints.
domain assumption Soft quadratic penalties on terminal position and velocity are sufficient to enforce the desired terminal speed without hard constraints
Required for the claim that weighting parameters alone can achieve arbitrary terminal speeds.

pith-pipeline@v0.9.1-grok · 5739 in / 1521 out tokens · 32258 ms · 2026-06-26T23:14:47.234591+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Lembeck, C. A., and Prussing, J. E., “Optimal impulsive i ntercept with low-thrust rendezvous return,” Journal of Guidance, Control, and Dynamics, Vol. 16, No. 3, 1993, pp. 426–433. https://doi.org/10.251 4/3.21027

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Zheng, Y

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work page doi:10.1016/j.ast 2020

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work page doi:10.2514/6.2026-2351 2026

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1998

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