A Riemannian Take on Distance Fields and Geodesic Flows in Robotics

TL;DR

The paper addresses the challenge of representing and exploiting spatial relationships on non-Euclidean robot manifolds by solving the Riemannian eikonal equation to obtain geodesic distance fields. It introduces NES, a grid-free, physics-informed neural network that learns a two-point distance field $U(\mathbf{q}_s, \mathbf{q}_e)$ and its gradient, enabling real-time geodesic backtracking on high-dimensional configuration spaces; the method is extensible via conditioning on task-space boundaries and on metric parameters $\omega$. NES leverages two loss terms, $\mathcal{L}_{\mathrm{eik}}$ and $\mathcal{L}_{\mathrm{div}}$, and enforces symmetry and non-negativity through a factorized distance representation, achieving globally consistent distance fields without labeled distances. The paper validates NES on kinetic-energy and Jacobi metrics, task-space pullbacks, and obstacle/stability scenarios, showing comparable or superior geodesic quality and substantial online efficiency relative to baselines like GRT and OC, with clear integration into QP-based control and task prioritization. Overall, the approach provides a scalable, geometry-aware prior that improves energy-efficient planning and control for both 2-DoF and 7-DoF manipulators, with potential to extend to broader metric spaces and dynamic programming frameworks.

Abstract

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.

Figures (20)

• Figure 1: Minimal distance paths as geodesics in the Euclidean space (a) and in another Riemannian metric space (b). The ellipses depict the SPD weighting matrices used to locally compute distances with this metric (isocontours of inverse matrices). A Riemannian manifold can be described intrinsically by the depicted metric. For visualization, it can also be depicted with corresponding extrinsic geometry in a higher-dimensional space (see inset), but geodesic computation does not require this construction and instead only requires the metric as an intrinsic geometry representation.
• Figure 2: Two approaches for computing geodesics on inhomogeneous manifolds. The traditional method (1) formulates the problem as an iterative solution to a differential equation, often using second-order optimization (e.g., Gauss-Newton path optimization). This approach requires a good initial guess and must be solved separately for each point pair. In contrast, we propose a wavefront propagation approach that first computes a geodesic distance field from a source point by solving the Riemannian eikonal equation (2a), then retrieves geodesics by backtracking along the gradient of this field (2b). We employ physics-informed neural networks (PINNs) to solve the eikonal equation, enabling scalable solutions in high-dimensional settings. This method encodes the manifold’s intrinsic geometry and yields globally optimal geodesics. It also offers modularity and efficiency: the distance field can be trained offline and used online for fast distance and geodesic queries.
• Figure 3: (a) Configuration space manifold endowed with a Riemannian metric using inertia as weighting matrix (visualized as isocontours of inverse matrices). The geodesics on this manifold correspond to minimal kinetic energy paths. By starting from a given point (red star), we can solve the eikonal equation on this manifold, accounting for the distance field (b) and gradient flow (c), which can then be used to backtrack geodesics in a very rapid manner (in milliseconds), see colored paths for examples of retrieved trajectories. Here, the source point is fixed for visualization. By using the proposed Neural Riemannian eikonal Solver (NES), these points are given as inputs, meaning that geodesics from any starting point to any final point are considered altogether. (d) Geodesic path (solid line) and Euclidean path (dashed line) on this manifold with corresponding robot motions.
• Figure 4: Solutions of Neural Riemannian Eikonal Solver (NES). (a) and (b) show the distance field and geodesic flow with the same parameters as Riemannian Fast Marching (RFM). (c) compares the difference with Figure \ref{['fig:RFM']} (b) (c), where geodesic flows produced by NES and RFM are shown in yellow and green, respectively. These three figures demonstrate that the neural network parameterization can solve the Riemannian eikonal equation, yielding results similar to those of RFM. (d) shows trajectories from source (green) to goal (orange) points and vice versa, highlighting the generalizability and symmetry of NES for arbitrary joint angle configuration pairs.
• Figure 5: Given a kinetic-energy metric, (a) and (b) show the distance field and geodesic flow for the target position $(2.0,2.0)$ in task space by using C-NES. Here, we do not specify the target joint angles (red stars). These joint angle targets are instead learned implicitly by the neural network. (c) shows four robot motions in task and configuration spaces (with different colors). We can observe that the motion solution for the task in orange color differs from the other three, which are automatically computed in accordance with the distance field and geodesic flow.