A Riemannian Take on Distance Fields and Geodesic Flows in Robotics

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.

💡 Research Summary

This paper addresses a fundamental limitation of conventional Euclidean signed distance fields (SDFs) in robotics: their inability to capture the intrinsic non‑Euclidean geometry of many robot configuration spaces. By formulating the problem as a Riemannian eikonal equation, the authors define a global distance field U(x) that satisfies ‖∇U(x)‖_{G^{-1}(x)} = 1, where G(x) is a smoothly varying symmetric positive‑definite (SPD) metric tensor encoding robot‑specific properties such as inertia, potential energy, or task‑space weighting. The solution U(x) provides not only the geodesic distance between any point and a source but also a gradient flow V(x) = G^{-1}(x)∇U(x) whose integral curves are the geodesics themselves. This perspective reframes geodesic computation from pairwise shooting or optimal‑control methods into a single‑source, globally consistent implicit representation.

To overcome the curse of dimensionality inherent in grid‑based solvers like Fast Marching Method (FMM), the authors propose a Neural Eikonal Solver (NES) built on physics‑informed neural networks (PINNs). NES learns a continuous mapping x ↦ U_θ(x) without any labeled distance data; the loss combines the PDE residual, the unit‑source boundary condition, and optional metric‑consistency terms. Automatic differentiation supplies exact gradients, enabling the network to satisfy the eikonal equation directly. Two variants are introduced: NES‑B, which conditions the network on source (or goal) coordinates to produce source‑specific distance fields, and NES‑M, which additionally takes the spatially varying metric G(x) as input, allowing a single model to handle environments where the metric changes across space.

The paper validates NES on several robotic scenarios. In a planar manipulator, a kinetic‑energy metric derived from the joint inertia matrix yields curved shortest paths that respect the robot’s dynamics. For a 7‑DoF Franka arm, a Jacobi metric (a conformal scaling of the inertia matrix by the remaining kinetic energy) produces geodesics that coincide with energy‑optimal trajectories under a fixed total energy constraint. Pullback metrics derived from task‑space objectives generate paths that naturally avoid obstacles and respect manipulability ellipsoids. Across all experiments, NES‑generated geodesics are shorter and consume less energy than baselines based on iterative shooting or gradient‑based path optimization, while query times are on the order of milliseconds. Moreover, because NES is mesh‑free, it scales to high‑dimensional configuration spaces with modest memory footprints, and training can be accelerated on GPUs using large batches of randomly sampled points.

Beyond path planning, the authors discuss how the continuous, differentiable distance field can be integrated into learning‑based control (e.g., shaping reinforcement‑learning rewards), constrained optimization (e.g., collision‑avoidance constraints expressed via distance gradients), and model‑predictive control pipelines. Limitations include potential convergence issues when the metric varies abruptly or when the underlying manifold has complex topology; the authors suggest future work on multi‑source fields, handling non‑symmetric metrics, and lightweight architectures for real‑time deployment on embedded robot hardware.

In summary, this work bridges Riemannian geometry and modern neural PDE solvers to provide a scalable, flexible, and real‑time capable framework for distance fields and geodesic flows in robotics. By learning the Riemannian eikonal equation in a self‑supervised manner, NES delivers globally optimal, energy‑aware paths without the need for precomputed distance labels, opening new avenues for geometry‑aware planning, control, and learning in high‑dimensional, non‑Euclidean robot spaces.