Physics-informed Neural Motion Planners (PiNMPs) 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.

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 paper, 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 paper introduces a new approach called Active Neural Time Fields (Active NTFields), 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 6 degrees-of-freedom (DOF) robot manipulator. The supplementary videos can be found at https://youtu.be/qTPL5a6pRKk, and the implementation code repository is available at https://github.com/Rtlyc/antfields-demo.

Motion Planning (MP) is a critical challenge in robotics, especially pertinent with the burgeoning interest in embodied artificial intelligence. Traditional MP methods often struggle with high-dimensional complexities. Recently neural motion planners, particularly physics-informed neural planners based on the Eikonal equation, have been proposed to overcome the curse of dimensionality. However, these methods perform poorly in complex scenarios with shaped robots due to multiple solutions inherent in the Eikonal equation. To address these issues, this paper presents PC-Planner, a novel physics-constrained self-supervised learning framework for robot motion planning with various shapes in complex environments. To this end, we propose several physical constraints, including monotonic and optimal constraints, to stabilize the training process of the neural network with the Eikonal equation. Additionally, we introduce a novel shape-aware distance field that considers the robot's shape for efficient collision checking and Ground Truth (GT) speed computation. This field reduces the computational intensity, and facilitates adaptive motion planning at test time. Experiments in diverse scenarios with different robots demonstrate the superiority of the proposed method in efficiency and robustness for robot motion planning, particularly in complex environments.

Motion planning (MP) is one of the core robotics problems requiring fast methods for finding a collision-free robot motion path connecting the given start and goal states. Neural motion planners (NMPs) demonstrate fast computational speed in finding path solutions but require a huge amount of expert trajectories for learning, thus adding a significant training computational load. In contrast, recent advancements have also led to a physics-informed NMP approach that directly solves the Eikonal equation for motion planning and does not require expert demonstrations for learning. However, experiments show that the physics-informed NMP approach performs poorly in complex environments and lacks scalability in multiple scenarios and high-dimensional real robot settings. To overcome these limitations, this paper presents a novel and tractable Eikonal equation formulation and introduces a new progressive learning strategy to train neural networks without expert data in complex, cluttered, multiple high-dimensional robot motion planning scenarios. The results demonstrate that our method outperforms state-of-the-art traditional MP, data-driven NMP, and physics-informed NMP methods by a significant margin in terms of computational planning speed, path quality, and success rates. We also show that our approach scales to multiple complex, cluttered scenarios and the real robot set up in a narrow passage environment. The proposed method's videos and code implementations are available at https://github.com/ruiqini/P-NTFields.

Neural Motion Planners (NMPs) have emerged as a promising tool for solving robot navigation tasks in complex environments. However, these methods often require expert data for learning, which limits their application to scenarios where data generation is time-consuming. Recent developments have also led to physicsinformed deep neural models capable of representing complex dynamical Partial Differential Equations (PDEs). Inspired by these developments, we propose Neural Time Fields (NTFields) for robot motion planning in cluttered scenarios. Our framework represents a wave propagation model generating continuous arrival time to find path solutions informed by a nonlinear first-order PDE called the Eikonal equation. We evaluate our method in various cluttered 3D environments, including the Gibson dataset, and demonstrate its ability to solve motion planning problems for 4-DOF and 6-DOF robot manipulators where the traditional grid-based Eikonal planners often face the curse of dimensionality. Furthermore, the results show that our method exhibits high success rates and significantly lower computational times than the state-of-the-art methods, including NMPs that require training data from classical planners. Our code is released: https://github.com/ruiqini/ Motion Planning (MP) is one of the core components of an autonomous robot system that aims to interact physically with its surrounding environments. MP algorithms find path solutions from the robot's start state to the goal state while respecting all constraints, such as collision avoidance. The quest for fast, scalable MP methods has led from traditional approaches such as RRT* (LaValle et al., 2001), Informed-RRT* (Gammell et al., 2014), and FMT* (Janson et al., 2015) to NMPs that exhibit promising performance in high-dimensional spaces.