Yiming Li*, Jiacheng Qiu* and Sylvain Calinon
Idiap Research Institute and EPFL
Abstract
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.
Riemannian Geometries in Robotics
Distance Fields and Geodesic Flows on Riemannian Manifolds
Riemannian metrics in the configuration space manifold enable the integration of many robot properties such as inertia, stiffness, or manipulability ellipsoids.
The distance field is a solution of the Riemannian eikonal equation, a first-order partial differential equation that simulates wavefront propagation on the manifold. Geodesics are tracked iteratively through gradient flow backpropagation.
Free Space
with Obstacles
Neural Riemannian Eikonal Solver
We present a neural network parameterization of the Riemannian eikonal equation, which is grid-free and scales better to high dimensions compared to numerical approaches. It also supports more efficient and flexible distance and gradient queries for arbitrary points.
It can also be extended to inverse kinematics, indicating how geodesic flows across task and joint spaces. Here the red star is the target position of the end-effector and the green stars are solutions learned implicitly by our approaches. Geodesic flows point to minimal-distance solutions on the Riemannian manifold.
Experiments
We demonstrate the approach in an energy-aware motion generation task, where robot motions occur on Riemannian manifolds defined by the robot's dynamics, such as kinetic energy and energy conservation. Geodesic motions lead to energy-efficient paths.
C-Space Motions
inverse Kinematics
7-axis Franka Robot
C-Space Motions
Inverse Kinematics
Task-prioritization