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：机器人领域的许多问题，例如从含噪声传感器数据估计状态或对齐两片LiDAR点云，均可表述为最小二乘问题并求解。然而，最小二乘问题的非最小化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（如伪Huber、Cauchy和Geman-McClure）以降低对异常值的敏感性。近年来，这些损失函数被统一为单一损失函数，可根据残差分布自适应地选择最优损失函数。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仅能在给定先验状态估计的情况下进行局部求解。本文的第一个贡献是将渐进非凸性（GNC）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）均基于高斯型分布。但残差通常被定义为多元误差的范数平方，呈类卡方分布。本文的第二个贡献是在GNC框架中引入了一种范数感知的自适应鲁棒损失函数。与最先进方法相比，该方法具有更强的鲁棒性。仿真与实验表明，与其他GNC公式相比，所提方法鲁棒性更强且收敛速度更快。