A common approach to localize a mobile robot is by measuring distances to points of known positions, called anchors. Locating a device from distance measurements is typically posed as a non-convex optimization problem, stemming from the nonlinearity of the measurement model. Non-convex optimization problems may yield suboptimal solutions when local iterative solvers such as Gauss-Newton are employed. In this paper, we design an optimality certificate for continuous-time range-only localization. Our formulation allows for the integration of a motion prior, which ensures smoothness of the solution and is crucial for localizing from only a few distance measurements. The proposed certificate comes at little additional cost since it has the same complexity as the sparse local solver itself: linear in the number of positions. We show, both in simulation and on real-world datasets, that the efficient local solver often finds the globally optimal solution (confirmed by our certificate), but it may converge to local solutions with high errors, which our certificate correctly detects.

翻译：移动机器人定位的常见方法是通过测量与已知位置点（称为锚点）之间的距离来实现。从距离测量值中确定设备位置通常被建模为一个非凸优化问题，其根源在于测量模型的非线性特性。当采用诸如高斯-牛顿法之类的局部迭代求解器时，非凸优化问题可能产生次优解。本文针对连续时间仅测距定位问题，设计了一种最优性认证方法。我们的公式允许集成运动先验，这确保了解的平滑性，并且对于仅通过少量距离测量值进行定位至关重要。所提出的认证方法计算成本极低，因为它与稀疏局部求解器具有相同的复杂度：与位置数量成线性关系。通过仿真和真实世界数据集，我们证明了高效的局部求解器通常能找到全局最优解（由我们的认证方法确认），但也可能收敛到高误差的局部解，我们的认证方法能够正确检测到这种情况。