Distance functions are crucial in robotics for representing spatial relationships between a robot and its environment. They provide an implicit, continuous, and differentiable representation that integrates seamlessly with control, optimization, and learning. While standard distance fields rely on the Euclidean metric, many robotic tasks inherently involve non-Euclidean structures. To this end, we generalize Euclidean distance fields to more general metric spaces by solving the Riemannian eikonal equation, a first-order partial differential equation whose solution defines a distance field and its associated gradient flow on the manifold, enabling the computation of geodesics and globally length-minimizing paths. We demonstrate that geodesic distance fields, the classical Riemannian distance function represented as a global, continuous, and queryable field, are effective for a broad class of robotic problems where Riemannian geometry naturally arises. To realize this, we present a neural Riemannian eikonal solver (NES) that solves the equation as a mesh-free implicit representation without grid discretization, scaling to high-dimensional robot manipulators. Training leverages a physics-informed neural network (PINN) objective that constrains spatial derivatives via the PDE residual and boundary and metric conditions, so the model is supervised by the governing equation and requires no labeled distances or geodesics. We propose two NES variants, conditioned on boundary data and on spatially varying Riemannian metrics, underscoring the flexibility of the neural parameterization. We validate the effectiveness of our approach through extensive examples, yielding minimal-length geodesics across diverse robot tasks involving Riemannian geometry.

翻译：距离函数在机器人学中对于表示机器人与其环境之间的空间关系至关重要。它们提供了一种隐式、连续且可微的表示，能够与控制、优化和学习无缝集成。虽然标准距离场依赖于欧几里得度量，但许多机器人任务本质上涉及非欧几里得结构。为此，我们通过求解黎曼程函方程，将欧几里得距离场推广到更一般的度量空间。该方程是一个一阶偏微分方程，其解定义了流形上的距离场及其相关的梯度流，从而能够计算测地线和全局长度最小化路径。我们证明，测地距离场——即表示为全局、连续且可查询场的经典黎曼距离函数——对于黎曼几何自然出现的一大类机器人问题是有效的。为实现这一点，我们提出了一种神经黎曼程函求解器（NES），它将该方程作为无网格的隐式表示进行求解，无需网格离散化，并可扩展到高维机器人机械臂。训练采用物理信息神经网络（PINN）目标，通过偏微分方程残差以及边界和度量条件来约束空间导数，因此模型受控于支配方程，无需标注的距离或测地线数据。我们提出了两种NES变体，分别以边界数据和空间变化的黎曼度量为条件，突显了神经参数化的灵活性。我们通过大量示例验证了所提方法的有效性，在涉及黎曼几何的各种机器人任务中生成了长度最小的测地线。