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).

Physics-informed Neural Motion Planners (PiN- MPs) provide a data-efficient framework for solving the Eikonal Partial Differential Equation (PDE) and representing the cost-to-go function for motion planning. However, their scalability remains limited by spectral bias and the complex loss landscape of PDE-driven training. Domain decomposition mitigates these issues by dividing the environment into smaller subdomains, but existing methods enforce continuity only at individual spatial points. While effective for function approximation, these methods fail to capture the spatial connectivity required for motion planning, where the cost- to-go function depends on both the start and goal coordinates rather than a single query point. We propose Finite Basis Neural Time Fields (FB-NTFields), a novel neural field representation for scalable cost-to-go estimation. Instead of enforcing continuity in output space, FB-NTFields construct a latent space representation, computing the cost-to-go as a distance between the latent embeddings of start and goal coordinates. This enables global spatial coherence while integrating domain decomposition, ensuring efficient large-scale motion planning. We validate FB-NTFields in complex synthetic and real-world scenarios, demonstrating substantial improvements over existing PiNMPs. Finally, we deploy our method on a Unitree B1 quadruped robot, successfully navigating indoor environments. The supplementary videos can be found at https://youtu.be/OpRuCbLNOwM.

Object manipulation skills are necessary for robots operating in various daily-life scenarios, ranging from warehouses to hospitals. They allow the robots to manipulate the given object to their desired arrangement in the cluttered environment. The existing approaches to solving object manipulations are either inefficient sampling based techniques, require expert demonstrations, or learn by trial and error, making them less ideal for practical scenarios. In this paper, we propose a novel, multimodal physics-informed neural network (PINN) for solving object manipulation tasks. Our approach efficiently learns to solve the Eikonal equation without expert data and finds object manipulation trajectories fast in complex, cluttered environments. Our method is multimodal as it also reactively replans the robot’s grasps during manipulation to achieve the desired object poses. We demonstrate our approach in both simulation and real-world scenarios and compare it against state-of-the-art baseline methods. The results indicate that our approach is effective across various objects, has efficient training compared to previous learning-based methods, and demonstrates high performance in planning time, trajectory length, and success rates. Our demonstration videos can be found at https://youtu.be/FaQLkTV9knI.

Offline Goal-Conditioned Reinforcement Learning (GCRL) holds great promise for domains such as autonomous navigation and locomotion, where collecting interactive data is costly and unsafe. However, it remains challenging in practice due to the need to learn from datasets with limited coverage of the state-action space and to generalize across long-horizon tasks. To improve on these challenges, we propose a Physics-informed (Pi) regularized loss for value learning, derived from the Eikonal Partial Differential Equation (PDE) and which induces a geometric inductive bias in the learned value function. Unlike generic gradient penalties that are primarily used to stabilize training, our formulation is grounded in continuous-time optimal control and encourages value functions to align with cost-to-go structures. The proposed regularizer is broadly compatible with temporal-difference-based value learning and can be integrated into existing Offline GCRL algorithms. When combined with Hierarchical Implicit Q-Learning (HIQL), the resulting method, Eikonal-regularized HIQL (Eik-HIQL), yields significant improvements in both performance and generalization, with pronounced gains in stitching regimes and large-scale navigation tasks. Code is available at https://github.com/VittorioGiammarino/Eik-HIQL.

Mapping and motion planning are two essential elements of robot intelligence that are interdependent in generating environment maps and navigating around obstacles. The existing mapping methods create maps that require computationally expensive motion planning tools to find a path solution. In this article, we propose a new mapping feature called arrival time fields, which is a solution to the Eikonal equation. The arrival time fields can directly guide the robot in navigating the given environments. Therefore, this article introduces a new approach called active neural time fields, which is a physics-informed neural framework that actively explores the unknown environment and maps its arrival time field on the fly for robot motion planning. Our method does not require any expert data for learning and uses neural networks to directly solve the Eikonal equation for arrival time field mapping and motion planning. We benchmark our approach against state-of-the-art mapping and motion planning methods and demonstrate its superior performance in both simulated and real-world environments with a differential drive robot and a six-degree-of-freedom robot manipulator.

Constrained Motion Planning (CMP) aims to find a collision-free path between the given start and goal configurations on the kinematic constraint manifolds. These problems appear in various scenarios ranging from object manipulation to legged-robot locomotion. However, the zero-volume nature of manifolds makes the CMP problem challenging, and the state-of-the-art methods still take several seconds to find a path and require a computationally expansive path dataset for imitation learning. Recently, physics-informed motion planning methods have emerged that directly solve the Eikonal equation through neural networks for motion planning and do not require expert demonstrations for learning. Inspired by these approaches, we propose the first physics-informed CMP framework that solves the Eikonal equation on the constraint manifolds and trains neural function for CMP without expert data. Our results show that the proposed approach efficiently solves various CMP problems in both simulation and real-world, including object manipulation under orientation constraints and door opening with a high-dimensional 6-DOF robot manipulator. In these complex settings, our method exhibits high success rates and finds paths in sub-seconds, which is many times faster than the state-of-the-art CMP methods.

Generating locally optimal UAV-trajectories is challenging due to the non-convex constraints of collision avoidance and actuation limits. We present the first local, optimization-based UAV-trajectory generator that simultane-ously guarantees validity and asymptotic optimality for known environments. Validity: Given a feasible initial guess, our algo-rithm guarantees the satisfaction of all constraints throughout the process of optimization. Asymptotic Optimality: We use an asymptotic exact piecewise approximation of the trajectory with an automatically adjustable resolution of its discretization. The trajectory converges under refinement to the first-order stationary point of the exact non-convex programming problem. Our method has additional practical advantages including joint optimality in terms of trajectory and time-allocation, and robustness to challenging environments as demonstrated in our experiments.

We propose a novel approach, calledProgressive Parameterizations, to compute foldover-free parameterizations with low isometric distortion on disk topology meshes. Instead of using the input mesh as a reference to define the objective function, we introduce a progressive reference that contains bounded distortion to the parameterized mesh and is as close as possible to the input mesh. After optimizing the bounded distortion energy between the progressive reference and the parameterized mesh, the parameterized mesh easily approaches the progressive reference, thereby also coming close to the input. By iteratively generating the progressive reference and optimizing the bounded distortion energy to update the parameterized mesh, our algorithm achieves high-quality parameterizations with strong practical reliability and high efficiency. We have demonstrated that our algorithm succeeds on a massive test data set containing over 20712 complex disk topology meshes. Compared to the state-of-the-art methods, our method has achieved higher computational efficiency and practical reliability.