Technical Summary: A Riemannian Take on Distance Fields and Geodesic Flows in Robotics

Problem Statement

In robotics, representing spatial relationships and planning motions often requires measuring distances in non-Euclidean spaces, such as the configuration space of a robot manipulator. While Signed Distance Fields (SDFs) are standard in Euclidean space for representing geometry and enabling efficient queries, they fail to capture the intrinsic geometric properties (e.g., inertia, stiffness, manipulability) that define robotic tasks.

Existing methods for computing geodesics (shortest paths) on Riemannian manifolds typically rely on:

1. Local Optimization: Solving boundary-value problems or shooting methods for specific start-goal pairs. These are computationally expensive, require good initial guesses, and lack a global representation of the manifold.
2. Grid-Based PDE Solvers: Methods like the Fast Marching Method (FMM) solve the eikonal equation on discretized grids. While they provide global solutions, they suffer from the "curse of dimensionality," making them intractable for high-degree-of-freedom (DoF) robots due to memory and resolution constraints.

The core challenge is to obtain a global, continuous, and differentiable distance field on high-dimensional Riemannian manifolds that encodes geodesic flows, enabling efficient, energy-aware motion planning without the scalability limitations of grid-based methods or the local-minima issues of shooting methods.

Methodology

The paper proposes a framework to solve the Riemannian eikonal equation, a first-order partial differential equation (PDE) that defines the distance field and its associated gradient flow on a manifold.

1. Mathematical Formulation

The authors generalize the isotropic eikonal equation to Riemannian manifolds. For a manifold $\mathcal{M}$ with a Riemannian metric tensor $G(x)$, the distance field $U(x)$ from a source $x_1$ satisfies:
$$\|\nabla U(x)\|_{G^{-1}(x)} = 1, \quad \text{s.t. } U(x_1) = 0$$
where $\|\cdot\|_{G^{-1}}$ is the norm induced by the inverse metric tensor. The geodesic flow $V(x)$, which allows for path backtracking, is defined as $V(x) = G^{-1}(x)\nabla U(x)$.

The paper utilizes various Riemannian metrics relevant to robotics:

• Kinetic-Energy Metric: $G(q) = M(q)$, where $M(q)$ is the inertia matrix. Geodesics here minimize kinetic energy.
• Jacobi Metric: $G(q) = 2(H - P(q))M(q)$, incorporating potential energy $P(q)$ and total energy $H$ for energy-conserving motions.
• Pullback Metric: Induced from task-space metrics (e.g., stiffness or manipulability) to the configuration space.
• Obstacle-Aware Metrics: Anisotropic metrics that penalize motion toward obstacles while allowing tangential movement.

2. Neural Riemannian Eikonal Solver (NES)

To overcome grid limitations, the authors introduce NES, a mesh-free solver based on Physics-Informed Neural Networks (PINNs).

• Architecture: A Multi-Layer Perceptron (MLP) takes source-goal pairs $(q_s, q_e)$ as input and outputs the predicted geodesic distance $U_\theta(q_s, q_e)$.
• Training Objective: The network is trained without labeled distance data. Instead, it minimizes a loss function derived from the Riemannian eikonal PDE residual:
  $$\mathcal{L}_{eik} = (\|\nabla_{q_e} U_\theta(q_s, q_e)\|_{G^{-1}(q_e)} - 1)^2$$
  A regularization term based on the Laplace-Beltrami operator is added to ensure smoothness and reduce gradient oscillations.
• Symmetry and Non-negativity: The network output is symmetrized ($U(q_s, q_e) = U(q_e, q_s)$) and factorized with a Euclidean term to ensure non-negativity and avoid singularities near the source.

3. Conditioned Variants (C-NES)

The authors propose two flexible variants of NES:

• Boundary-Conditioned: The network is conditioned on a task-space source $x_s$ rather than a joint-space source. The boundary condition is implicitly defined by the forward kinematics $f(q) = x_s$, allowing the solver to handle inverse kinematics (IK) and find the geodesically closest IK solution without explicit enumeration.
• Metric-Conditioned: The network accepts a parameter $\omega$ (e.g., total energy $H$) that modulates the Riemannian metric $G(q; \omega)$. This allows a single network to represent a family of distance fields for different dynamic regimes.

4. Control Integration

The learned distance field provides a global policy. To track the geodesic flow in real-time, the authors blend the learned gradient direction with a local linear attraction term. This desired velocity is fed into standard controllers:

• Quadratic Programming (QP): Kinematic (velocity) or kinodynamic (torque) QPs track the desired velocity while respecting joint limits and dynamics.
• Task Prioritization: The geodesic flow is projected into the null-space of higher-priority constraints (e.g., obstacle avoidance or end-effector orientation), ensuring strict constraint satisfaction while optimizing energy efficiency in the remaining degrees of freedom.

Key Contributions

1. Riemannian Eikonal Formulation: Reframing geodesic computation as a single global implicit representation of the manifold, enabling fast backtracking and direct integration with planning/control, distinct from iterative shooting or optimal control methods.
2. Neural Riemannian Eikonal Solver (NES): A grid-free, differentiable solver that scales to high-dimensional configuration spaces (e.g., 7-DoF arms) by learning the PDE solution via PINNs, requiring no labeled distance data.
3. Conditioned Variants: Introduction of C-NES variants that condition on boundary data (handling IK implicitly) and spatially varying metrics (handling dynamic energy regimes), demonstrating the flexibility of neural parameterization.
4. Empirical Validation: Extensive experiments across kinematics, dynamics, and motion planning, showing that NES generates geodesics that align with energy-efficient trajectories and outperform baselines in high-dimensional settings.

Results

The authors evaluated NES on 2-DoF planar and 7-DoF Franka manipulators against baselines including Geodesic Ray Tracing (GRT), Supervised Learning (SL), Geodesic Shooting (GS), Optimal Control (OC), and Riemannian Fast Marching (RFM).

• Energy Efficiency: On energy-aware manifolds (Jacobi metric), NES-generated paths consistently achieved lower active torque and total control effort compared to GS and OC. In 7-DoF experiments, NES produced the shortest geodesics and lowest active torques among all methods.
• Scalability and Speed:
  • Offline: Training takes ~5 minutes for 2-DoF and ~4 hours for 7-DoF.
  • Online: Inference is extremely fast (milliseconds), supporting real-time control. Unlike RFM, which must recompute for every new source, NES generalizes to arbitrary start-goal pairs instantly.
  • Robustness: NES avoids local minima traps common in Optimal Control and Geodesic Shooting, particularly for long-range tasks.
• Task-Space Integration: C-NES successfully mapped task-space targets to joint-space configurations, implicitly solving IK and selecting energy-optimal solutions among redundant configurations.
• Constraint Handling: When integrated with task-prioritized control, NES maintained energy efficiency while strictly satisfying constraints like obstacle avoidance and end-effector orientation.

Significance and Claims

The paper claims that solving the Riemannian eikonal equation via neural networks provides a principled, scalable, and flexible geometric prior for robotics.

• Global vs. Local: Unlike local optimization methods that require re-solving for every query, NES provides a global distance field that captures the manifold's structure, enabling rapid, consistent queries.
• Physics-Informed Learning: By training on the PDE residual rather than labeled data, the method avoids the computational cost of generating ground-truth geodesics and ensures the solution adheres to the underlying geometric laws.
• Dynamics-Aware Planning: The approach naturally incorporates system dynamics (inertia, potential energy) into the path planning process, yielding trajectories that are not just geometrically short but also energetically efficient.
• Modularity: The framework decouples planning (finding the geodesic flow) from control (tracking the flow), allowing the use of standard QP controllers while leveraging global geometric optimality.

The authors acknowledge that while the method is theoretically grounded, training the neural solver on highly anisotropic metrics can be challenging. They position the approach as a heuristic that provides globally optimal paths for analyzing and planning robot motion, particularly where both kinetic and potential energy are modeled, rather than a perfect substitute for all actuator cost models.