Topological Control and Orbital Steady-State Analysis of Megascale LEO Constellations Based on Discrete Order Geometry (DOG)

Author: Zhang Suhang

Affiliation: Luoyang School of Mathematics

Abstract

With megascale low-Earth orbit (LEO) constellations represented by Starlink expanding beyond the ten-thousand-satellite threshold, traditional orbital mechanics modelling and network routing algorithms suffer severely from the curse of dimensionality and computational bottlenecks. Conventional approaches treat satellites as mass points in a continuous space, where high-complexity N-body perturbation calculations and real-time inter-satellite distance resolution consume massive computing resources. To address this challenge, this paper introduces Discrete Order Geometry (DOG) theory and constructs a novel constellation topological model. Within this framework, each individual satellite is defined as a lattice element embedded with a local continuous dynamical manifold. Inter-satellite communication and orbital constraints are abstracted into discrete hierarchical partial-order relations via the Zhang Matrix, and global steady-state control is realised through the Fundamental Coupling Equation (FCE). The study demonstrates that this framework fully preserves the continuous orbital mechanical properties of single satellites at the micro level, while transforming the constellation network into a sparse discrete dynamical system at the macro level. It drastically reduces the computational complexity of system reconfiguration, providing a robust underlying mathematical framework for collision avoidance, inter-satellite laser communication routing and local fault tolerance of future ten-thousand-satellite constellations.

Keywords: Starlink; megaconstellation; Discrete Order Geometry (DOG); Zhang Matrix; orbital steady state; network topology

1 Introduction

Commercial aerospace and LEO communication networks are rapidly evolving toward ultra-large-scale constellations. For thousands of high-speed orbiting satellites, traditional modelling based on continuous spatial coordinates encounters formidable challenges. On one hand, high-precision continuous orbital integration accounting for Earth’s non-spherical gravity, atmospheric drag, and lunar-solar three-body perturbations incurs heavy computational costs. On the other hand, the topological structure of inter-satellite links (ISLs) fluctuates drastically with the relative motion of satellites, rendering routing algorithms extremely intricate.

Essentially, a megaconstellation constitutes a strongly coupled system characterised by continuity at the microscopic level (single-satellite orbit) and discreteness at the macroscopic level (network topology). Traditional mathematical models tend to perform unified calculations within a globally continuous framework, thereby falling into a computational bottleneck. Based on Discrete Order Geometry (DOG) proposed by the Luoyang School of Mathematics, this paper reconstructs the constellation network as a hierarchically nested discrete geometric space. The objective is to achieve global constellation topological stability and efficient control at substantially reduced computational expense.

2 DOG Geometric Modelling of the Constellation System

The construction of the megaconstellation system under the DOG framework follows the principle of hierarchical separation.

2.1 Definition of Lattice Element: Local Continuous Manifold for Individual Satellites

Within DOG, each satellite is defined as an independent lattice element. Unlike conventional zero-dimensional mass points, this lattice element is equipped with a complete Riemannian manifold structure. Accordingly, the orbital dynamics and attitude dynamics of a single satellite, including continuous thrust variation and antenna pointing angles, are entirely encapsulated inside the lattice element. Classical orbital mechanics remains fully valid within the lattice element and is unaffected by macroscopic discretisation.

2.2 Zhang Matrix: Discrete Coupling Network for Constellation Topology

No continuous spatial distance linkage is imposed between satellites. Instead, discrete hierarchical partial-order relations are established via the Zhang Matrix.

- Sparsity: The Zhang Matrix is generated by continued-fraction recursion. Non-zero coupling weights are assigned only to satellites satisfying specific elevation-angle criteria, or satellites within the same or adjacent orbital planes.

- Dynamic reconfiguration: When relative satellite motion occurs, only the weights or sparse structure of the Zhang Matrix require updating. There is no need to recompute continuous relative distances for all satellites across the whole system, which substantially cuts the overhead of routing switching.

3 Steady-State Control of Orbit and Network via the FCE

To guarantee operational stability during constellation service, the Fundamental Coupling Equation (FCE) is adopted as the global constraint:

\lambda_p\,\omega_p \prod_{q\in\text{Neigh}(p)} C_{pq}\,\omega_q = 1

where \omega_p denotes the orbital state or communication band coefficient of the p-th satellite; C_{pq} stands for the coupling weight between adjacent satellites in the Zhang Matrix.

3.1 Collision Avoidance and Orbital Station-Keeping

When a satellite deviates from its nominal orbit due to perturbations such as solar wind or thrust anomaly, the local partial-order relation is altered. The multiplicative constraint of the FCE drives minor adjustments to the orbital parameters of neighbouring satellites in \text{Neigh}(p), enabling rapid convergence of the whole system to a new steady-state solution and achieving autonomous collision avoidance and orbital self-healing for the constellation.

3.2 Optimisation of Inter-Satellite Communication Routing

For inter-satellite laser communication networks, the multiplicative relation of the FCE naturally maps the signal power and bandwidth attenuation law within the hierarchical network. Provided the steady-state solution of the FCE is maintained, even at the scale of ten thousand satellites, the sparse hierarchical structure of the Zhang Matrix can be leveraged to rapidly identify the shortest and most stable communication path.

4 Engineering Advantages and Application Prospects

1. Computational dimensionality reduction: Global N-body continuous computation is reduced to local Riemannian manifold updates and sparse matrix operations, conserving computing resources on both on-board computers and ground control centres.

2. High robustness: If a satellite fails or becomes decommissioned, the corresponding lattice element is simply removed from the Zhang Matrix to sever its partial-order connections. Since the system relies on a global sparse structure, a local failure will not trigger cascading global collapse, and the constellation can complete topological reconfiguration instantaneously.

​

3. Multi-scale compatibility: The framework is not limited to Starlink-type LEO constellations. It can be seamlessly extended to hybrid constellations comprising LEO, MEO and GEO spacecraft, with different orbital altitudes treated as macroscopic hierarchical layers within DOG.

5 Conclusion

This paper proposes a novel topological control and steady-state analysis framework for megascale LEO constellations built upon Discrete Order Geometry (DOG). Continuous dynamics are confined inside each satellite lattice element, while discrete topological relations are defined between lattice elements. This approach fundamentally resolves the curse of dimensionality arising from large-scale continuous computation in traditional mechanics. This work represents a practical engineering implementation of the mathematical resolution of the discrete–continuum dichotomy. It establishes an original underlying mathematical and control paradigm for future ten-thousand-satellite space networks.