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.

翻译：估计局部系统相对于全球导航卫星系统（GNSS）参考系的绝对姿态，常受限于局部极小值问题，且高度依赖于卫星可见性。现有的对齐方法要么依赖于在GNSS信号受限环境中难以获得的充足卫星数量，要么采用无法保证解最优性的局部优化方法。本文提出了一种全局最优求解器，将原始伪距或多普勒测量值转化为凸松弛问题。所提方法是可认证的，意味着能够数值验证结果的正确性，填补了现有局部优化器无法保证最优性的空白。我们首先将原始坐标系对齐问题表述为一个非凸二次约束二次规划问题，并将该QCQP问题松弛为一个凹拉格朗日对偶问题，该对偶问题为原问题提供了下界。随后，通过松弛紧致性和可观测性分析，推导出解可认证最优性的判定准则。最后，通过仿真和真实世界实验对所提方法进行评估。实验表明，即使在仅有2颗卫星提供多普勒测量且车辆进行二维运动的情况下，我们的方法仍能提供可认证的最优解；而传统的基于速度的VOBA方法以及先进的GVINS对齐技术则可能失败或在无提示的情况下收敛至局部最优解。为支持机器人领域基于GNSS的导航技术发展，所有代码与数据均已开源：https://github.com/Baoshan-Song/Certifiable-Doppler-alignment。