Physics-informed Temporal Difference Metric Learning for Robot Motion Planning

TL;DR

This work tackles robust robot motion planning by reframing the Eikonal equation as the optimal value function and geodesic-distance on a Riemannian manifold, enabling self-supervised learning without expert data. The authors introduce a temporal-difference metric-learning framework that combines an infinitesimal Eikonal loss with a finite-step TD loss, obstacle-normal alignment, and a causality prior to enhance global consistency and convergence. A metric-learning architecture with environment-conditioned attention preserves geodesic properties while enabling generalization to unseen scenes, and a sampling-based MPC routinely infers paths from the learned travel-time field. Empirical results across 2–12 DOF tasks show improved accuracy, scalability, and generalization over prior self-supervised methods, with notably fast planning times and strong sim2real performance in high-dimensional setups. The approach offers a practical, scalable avenue for robust motion planning in complex, cluttered environments.

Abstract

The motion planning problem involves finding a collision-free path from a robot's starting to its target configuration. Recently, self-supervised learning methods have emerged to tackle motion planning problems without requiring expensive expert demonstrations. They solve the Eikonal equation for training neural networks and lead to efficient solutions. However, these methods struggle in complex environments because they fail to maintain key properties of the Eikonal equation, such as optimal value functions and geodesic distances. To overcome these limitations, we propose a novel self-supervised temporal difference metric learning approach that solves the Eikonal equation more accurately and enhances performance in solving complex and unseen planning tasks. Our method enforces Bellman's principle of optimality over finite regions, using temporal difference learning to avoid spurious local minima while incorporating metric learning to preserve the Eikonal equation's essential geodesic properties. We demonstrate that our approach significantly outperforms existing self-supervised learning methods in handling complex environments and generalizing to unseen environments, with robot configurations ranging from 2 to 12 degrees of freedom (DOF).

Figures (8)

• Figure 1: The plots depict the solution to the Eikonal equation $|\nabla_q T(q)| = 1, T(0) = 0$ with sampled green points as training data. In the top plot, the desired solution $T(q) = |q|$ is shown. In contrast, the bottom plot demonstrates that the green points also satisfy $|\nabla_q T(q)| = 1$, but without proper constraints, the solution deviates. The TD loss $L_{TD}$ ensures correctness by enforcing, for example, that for $q_1 > 0$, $T(q_1) - T(q_1 - d) = d$. Therefore, $L_{TD}$ is more capable of finding the ground truth solution.
• Figure 2: The figure shows the geodesics of points $A$ and $B$ on a circle under different embeddings. With $L_2$, the distance collapses into a line, causing overlap and ambiguity. In contrast, $L_1$ transforms it into a diamond, preserving the geodesic structure and resolving ambiguity.
• Figure 3: First row: We compare our method with FMM, NTF, and P-NTF. Middle row: We ablate our method by removing $-L_E$, $-L_{TD}$, $-L_N$, and $-L_C$. Third row: We replace our distance metric $D$ with IQE, PQE, MRN, and DN.
• Figure 4: Depiction of our (a) Gibson, (b) Cluttered 3D (C3D), and (c) 7-DOF Manipulator environments. We also illustrate multiple trajectories planned by our method between different start and goal pairs. It can be seen that our method finds smooth trajectories while avoiding collisions.
• Figure 5: Demontration of a path planned by our method to navigate real-world cabinet environment using a 12-DOF dual-arm robot. This particular trajectory was planned in 0.11 seconds.