-- 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.

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).

Geodesic distances play a fundamental role in robotics, as they efficiently encode global geometric information of the domain. Recent methods use neural networks to approximate geodesic distances by solving the Eikonal equation through physics-informed approaches. While effective, these approaches often suffer from unstable convergence during training in complex environments. We propose a framework to learn geodesic distances in irregular domains by using the Soner boundary condition, and systematically evaluate the impact of data losses on training stability and solution accuracy. Our experiments demonstrate that incorporating data losses significantly improves convergence robustness, reducing training instabilities and sensitivity to initialization. These findings suggest that hybrid data-physics approaches can effectively enhance the reliability of learning-based geodesic distance solvers with sparse data.

Distance functions are crucial in robotics for representing spatial relationships between the robot and the environment. It provides an implicit representation of continuous and differentiable shapes, which can seamlessly be combined with control, optimization, and learning techniques. While standard distance fields rely on the Euclidean metric, many robotic tasks inherently involve non-Euclidean structures. To this end, we generalize the use of Euclidean distance fields to more general metric spaces by solving a 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 show that this \emph{geodesic distance field} can also be exploited in the robot configuration space. To realize this concept, we exploit physics-informed neural networks to solve the eikonal equation for high-dimensional spaces, which provides a flexible and scalable representation without the need for discretization. Furthermore, a variant of our neural eikonal solver is introduced, which enables the gradient flow to march across both task and configuration spaces. As an example of application, we validate the proposed approach in an energy-aware motion generation task. This is achieved by considering a manifold defined by a Riemannian metric in configuration space, effectively taking the property of the robot's dynamics into account. Our approach produces minimal-energy trajectories for a 7-axis Franka robot by iteratively tracking geodesics through gradient flow backpropagation.

Existing losses such as the eikonal loss is to ensure that the implicit function indeed outputs cannot guarantee the recovered implicit function to be a the signed distance. A standard regularization loss used is distance function, even when the implicit function satisfies the eikonal equation: it constrains the norm of the gradient the eikonal equation almost everywhere. Furthermore, the of an implicit function to be 1 almost everywhere. If eikonal loss suffers from stability issues in optimization and the implicit function is a signed distance function, then it the remedies that introduce area or divergence minimization satisfies the eikonal equation. However, the converse is not can lead to oversmoothing. We address these challenges true. Figure 1 shows an example: on the left, we optimize by designing a loss function that when minimized can an implicit function to satisfy the eikonal equation, while converge to the true distance function, is stable, and naturally it successfully does so, it converges to a solution that is far penalize large surface area. We provide theoretical from the actual distance [5, 6].

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.

Computing distances on Riemannian manifolds is a challenging problem with numerous applications, from physics, through statistics, to machine learning. In this paper, we introduce the metric-constrained Eikonal solver to obtain continuous, differentiable representations of distance functions (geodesics) on manifolds. The differentiable nature of these representations allows for the direct computation of globally length-minimising paths on the manifold. We showcase the use of metric-constrained Eikonal solvers for a range of manifolds and demonstrate the applications. First, we demonstrate that metric-constrained Eikonal solvers can be used to obtain the Fréchet mean on a manifold, employing the definition of a Gaussian mixture model, which has an analytical solution to verify the numerical results. Second, we demonstrate how the obtained distance function can be used to conduct unsupervised clustering on the manifold - a task for which existing approaches are computationally prohibitive. This work opens opportunities for distance computations on manifolds.