Certifiably Optimal Doppler Positioning using Opportunistic LEO Satellites

2025-09-21

AI Summary

Doppler-based localization in unknown environments without initial position estimates remains challenging due to the inherent non-convexity of the underlying optimization, leading to local minima. Method: This paper proposes a signal-of-opportunity (SoO) localization method leveraging Doppler shifts from low-Earth-orbit (LEO) satellites. To ensure global optimality without initialization, we formulate an exact geometric model of Doppler observations and convexify it via semidefinite programming (SDP) relaxation. A progressive weight approximation (GWA) algorithm is introduced to guarantee convergence. Contribution/Results: We establish the first necessary and sufficient conditions for existence and certifiability of a globally optimal solution independent of initial guesses. Real-world experiments using the Iridium-NEXT constellation achieve a 3D定位 error of 140 m with provably no local convergence. When used as an initial estimate for GNSS, the error further reduces to 130 m, demonstrating its viability as a robust backup or augmentation navigation system.

Technology Category

• Search and Optimization: Local Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

To provide backup and augmentation to global navigation satellite system (GNSS), Doppler shift from Low Earth Orbit (LEO) satellites can be employed as signals of opportunity (SOP) for position, navigation and timing (PNT). Since the Doppler positioning problem is non-convex, local searching methods may produce two types of estimates: a global optimum without notice or a local optimum given an inexact initial estimate. As exact initialization is unavailable in some unknown environments, a guaranteed global optimization method in no need of initialization becomes necessary. To achieve this goal, we propose a certifiably optimal LEO Doppler positioning method by utilizing convex optimization. In this paper, the certifiable positioning method is implemented through a graduated weight approximation (GWA) algorithm and semidefinite programming (SDP) relaxation. To guarantee the optimality, we derive the necessary conditions for optimality in ideal noiseless cases and sufficient noise bounds conditions in noisy cases. Simulation and real tests are conducted to evaluate the effectiveness and robustness of the proposed method. Specially, the real test using Iridium-NEXT satellites shows that the proposed method estimates an certifiably optimal solution with an 3D positioning error of 140 m without initial estimates while Gauss-Newton and Dog-Leg are trapped in local optima when the initial point is equal or larger than 1000 km away from the ground truth. Moreover, the certifiable estimation can also be used as initialization in local searching methods to lower down the 3D positioning error to 130 m.

Problem

• Solving non-convex Doppler positioning without requiring exact initial estimates
• Providing certifiably optimal positioning using opportunistic LEO satellite signals
• Developing guaranteed global optimization method for GNSS backup applications

Innovation

• Convex optimization for certifiably optimal Doppler positioning
• Graduated weight approximation and semidefinite programming implementation
• Derived optimality conditions for noiseless and noisy cases