Certifiable Alignment of GNSS and Local Frames via Lagrangian Duality

TL;DR

This paper presents a globally optimal, certifiable solver for aligning local frames to GNSS references using convex relaxation of the problem, effective even with minimal satellite data.

Contribution

It introduces a novel convex relaxation approach for frame alignment that guarantees optimality verification, addressing limitations of existing local optimization methods.

Findings

The method achieves certifiably optimal solutions with as few as 2 satellites.

It outperforms traditional methods that may fail or get stuck in local optima.

Experiments validate effectiveness in both simulation and real-world scenarios.

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.