Many problems in robotics, such as estimating the state from noisy sensor data or aligning two LiDAR point clouds, can be posed and solved as least-squares problems. Unfortunately, vanilla nonminimal solvers for least-squares problems are notoriously sensitive to outliers. As such, various robust loss functions have been proposed to reduce the sensitivity to outliers. Examples of loss functions include pseudo-Huber, Cauchy, and Geman-McClure. Recently, these loss functions have been generalized into a single loss function that enables the best loss function to be found adaptively based on the distribution of the residuals. However, even with the generalized robust loss function, most nonminimal solvers can only be solved locally given a prior state estimate due to the nonconvexity of the problem. The first contribution of this paper is to combine graduated nonconvexity (GNC) with the generalized robust loss function to solve least-squares problems without a prior state estimate and without the need to specify a loss function. Moreover, existing loss functions, including the generalized loss function, are based on Gaussian-like distribution. However, residuals are often defined as the squared norm of a multivariate error and distributed in a Chi-like fashion. The second contribution of this paper is to apply a norm-aware adaptive robust loss function within a GNC framework. This leads to additional robustness when compared with state-of-the-art methods. Simulations and experiments demonstrate that the proposed approach is more robust and yields faster convergence times compared to other GNC formulations.

翻译：机器人学中的许多问题，例如从带噪声的传感器数据估计状态或对齐两幅激光雷达点云，均可构建并求解为最小二乘问题。遗憾的是，针对最小二乘问题的普通非最小化求解器极易受外点影响。为此，学者们提出了多种鲁棒损失函数以降低对外点的敏感性，例如伪Huber、柯西和Geman-McClure损失。近期，这些损失函数被统一为单一损失函数形式，使得能够根据残差分布自适应选择最优损失函数。然而，即使采用广义鲁棒损失函数，由于问题的非凸性，大多数非最小化求解器只能在给定先验状态估计的前提下进行局部求解。本文的第一项贡献是将渐进非凸性与广义鲁棒损失函数相结合，无需先验状态估计且无需指定损失函数即可求解最小二乘问题。此外，现有损失函数（包括广义损失函数）均基于类高斯分布。但残差通常被定义为多元误差的平方范数，呈现卡方分布特征。本文的第二项贡献是在渐进非凸性框架内应用一种范数感知自适应鲁棒损失函数。与现有先进方法相比，该方法具有更强的鲁棒性。仿真与实验表明，相较于其他渐进非凸性求解方案，本文提出的方法更为鲁棒且收敛速度更快。