Distance functions are crucial in robotics for representing spatial relationships between the robot and the environment. It provides an implicit representation of continuous and differentiable shapes, which can seamlessly be combined with control, optimization, and learning techniques. While standard distance fields rely on the Euclidean metric, many robotic tasks inherently involve non-Euclidean structures. To this end, we generalize the use of Euclidean distance fields to more general metric spaces by solving a 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 show that this \emph{geodesic distance field} can also be exploited in the robot configuration space. To realize this concept, we exploit physics-informed neural networks to solve the eikonal equation for high-dimensional spaces, which provides a flexible and scalable representation without the need for discretization. Furthermore, a variant of our neural eikonal solver is introduced, which enables the gradient flow to march across both task and configuration spaces. As an example of application, we validate the proposed approach in an energy-aware motion generation task. This is achieved by considering a manifold defined by a Riemannian metric in configuration space, effectively taking the property of the robot's dynamics into account. Our approach produces minimal-energy trajectories for a 7-axis Franka robot by iteratively tracking geodesics through gradient flow backpropagation.

翻译：距离函数在机器人学中对于表示机器人与环境之间的空间关系至关重要。它提供了连续可微形状的隐式表示，能够与控制、优化及学习技术无缝结合。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——一个一阶偏微分方程——将欧几里得距离场的应用推广至更一般的度量空间。该方程的解定义了流形上的距离场及其关联的梯度流，从而能够计算测地线与全局长度最小化路径。我们证明这种**测地距离场**同样可应用于机器人构型空间。为实现这一概念，我们利用物理信息神经网络求解高维空间的程函方程，提供了一种无需离散化的灵活可扩展表示。此外，我们引入了一种神经程函求解器的变体，使梯度流能够在任务空间与构型空间中同步推进。作为应用示例，我们在能量感知运动生成任务中验证了所提方法。该方法通过在构型空间中定义黎曼度量流形，有效考虑了机器人动力学特性。我们通过梯度流反向传播迭代追踪测地线，为七轴Franka机器人生成了最小能量轨迹。