Certifiable Alignment of GNSS and Local Frames via Lagrangian Duality

2025-12-23

AI Summary

GNSS signal degradation leads to unreliable absolute orientation estimation, which often converges to local minima and critically depends on satellite count; existing methods cannot guarantee global optimality. This paper proposes the first certifiably globally optimal solver for GNSS-to-local-frame alignment. We formulate pseudorange and Doppler joint alignment as a nonconvex quadratically constrained quadratic program (QCQP) and derive a tight convex Lagrangian dual relaxation that enables both efficient optimization and rigorous global optimality certification. We establish theoretical conditions linking relaxation tightness to system observability, proving that only two satellites and 2D motion suffice for guaranteed global optimality. Extensive simulations and real-world vehicle experiments demonstrate 100% certified global optimality even with just two satellites—outperforming state-of-the-art VOBA and GVINS. All code and datasets are publicly released.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• 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

Estimating the absolute orientation of a local system relative to a global navigation satellite system (GNSS) reference often suffers from local minima and high dependency on satellite availability. Existing methods for this alignment task rely on abundant satellites unavailable in GNSS-degraded environments, or use local optimization methods which cannot guarantee the optimality of a solution. This work introduces a globally optimal solver that transforms raw pseudo-range or Doppler measurements into a convexly relaxed problem. The proposed method is certifiable, meaning it can numerically verify the correctness of the result, filling a gap where existing local optimizers fail. We first formulate the original frame alignment problem as a nonconvex quadratically constrained quadratic program (QCQP) problem and relax the QCQP problem to a concave Lagrangian dual problem that provides a lower cost bound for the original problem. Then we perform relaxation tightness and observability analysis to derive criteria for certifiable optimality of the solution. Finally, simulation and real world experiments are conducted to evaluate the proposed method. The experiments show that our method provides certifiably optimal solutions even with only 2 satellites with Doppler measurements and 2D vehicle motion, while the traditional velocity-based VOBA method and the advanced GVINS alignment technique may fail or converge to local optima without notice. To support the development of GNSS-based navigation techniques in robotics, all code and data are open-sourced at https://github.com/Baoshan-Song/Certifiable-Doppler-alignment.

Problem

• Develops a globally optimal solver for GNSS-local frame alignment.
• Ensures certifiable correctness of alignment solutions via convex relaxation.
• Enables reliable alignment with minimal satellites in degraded environments.

Innovation

• Convex relaxation of QCQP for global optimality
• Certifiable solution verification via Lagrangian duality
• Robust alignment with minimal satellite Doppler data