Many problems in robotics, such as estimating the state from noisy sensor data or aligning two 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. The proposed approach enables a GNC formulation of a generalized loss function such that GNC can be readily applied to a wider family of loss functions. Furthermore, simulations and experiments demonstrate that the proposed method is more robust compared to non-GNC counterparts, and yields faster convergence times compared to other GNC formulations.

翻译：机器人学中的许多问题，例如从含噪传感器数据中估计状态或对齐两个点云，可表述为最小二乘问题求解。然而，标准的非最小化最小二乘求解器对异常值极为敏感。为此，研究者提出了多种鲁棒损失函数来降低异常值敏感性，例如伪Huber、Cauchy和Geman-McClure损失函数。近年来，这些损失函数被统一为一种通用损失函数，能够根据残差分布自适应地确定最优损失函数。然而，即使采用通用鲁棒损失函数，由于问题的非凸性，大多数非最小化求解器仅能在给定先验状态估计的前提下进行局部求解。本文的第一个贡献在于将渐进凸化（GNC）与通用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（包括通用损失函数）均基于类高斯分布，但残差通常定义为多元误差的平方范数，服从类卡方分布。本文的第二个贡献是在GNC框架中引入一种范数感知的自适应鲁棒损失函数。所提方法实现了通用损失函数的GNC形式化，使GNC能广泛适用于更多损失函数族。仿真与实验表明，与非GNC方法相比，所提方法具有更强的鲁棒性，且相比其他GNC形式化方法收敛速度更快。