Adaptive Distance Functions via Kelvin Transformation

TL;DR

This work introduces a semantics-aware distance function $h$ to encode task-specific safety in contact-rich robotic manipulation, enabling safe sets that reflect object affordances. The authors compute $h$ by solving a Laplace equation on the exterior of an object via the Kelvin Transformation, transforming the unbounded domain to a bounded one and obtaining exact boundary behavior using a tetrahedral mesh. The approach delivers sub-millisecond query times and online adaptability (e.g., updating with changing semantic states) while maintaining differentiability, suitable for real-time safety-critical control and teleoperation. Validation on a wrench demonstrates geometry-agnostic performance and practical viability, though the method requires known geometry and yields piecewise-smooth FEM solutions, suggesting future extensions to unknown objects and analytical-efficiency alternatives.

Abstract

The term safety in robotics is often understood as a synonym for avoidance. Although this perspective has led to progress in path planning and reactive control, a generalization of this perspective is necessary to include task semantics relevant to contact-rich manipulation tasks, especially during teleoperation and to ensure the safety of learned policies. We introduce the semantics-aware distance function and a corresponding computational method based on the Kelvin Transformation. This allows us to compute smooth distance approximations in an unbounded domain by instead solving a Laplace equation in a bounded domain. The semantics-aware distance generalizes signed distance functions by allowing the zero level set to lie inside of the object in regions where contact is allowed, effectively incorporating task semantics, such as object affordances, in an adaptive implicit representation of safe sets. In numerical experiments we show the computational viability of our method for real applications and visualize the computed function on a wrench with various semantic regions.

Figures (5)

• Figure 1: We implicitly define safe sets that consider the semantics of contact regions by introducing the semantics-aware distance, $h$. Its zero level set, $h(\mathbf{x})=0$, lies within the object, where contact with the surface is allowed ($\sigma>0$). We visualize the semantics-aware distance of a knife through its iso-surfaces -- starting at the zero level set -- which describes the safe set boundary. With our proposed computation method, query times are lower than $1µs$ and the function is continuously differentiable and is therefore useful for example in combination with control barrier function approaches to safety critical control.
• Figure 2: The computational method with the Kelvin Transformation approach is visualized in this figure, by using the 5th link of the Franka Robot (i) as an example object. The center of inversion is depicted as a red dot in insets (ii-iv). The inset (ii) shows the surface mesh of the object, (iii) shows the mesh transformed by the inversion map \ref{['eq:inversion_map']}, (iv) shows the generated interior tetrahedral mesh, and (v) shows an iso-surface representation of the solution with constant boundary values. Corresponding numerical statistics are shown in Table \ref{['tab:wrench_stats']}.
• Figure 3: Visualization of the tested mesh range, as parametrized by the exterior domain length $l_x$. Top row: $l_x=0.2$, bottom row: $l_x=0.02$. The vertices are projected to the exterior domain through the inversion map \ref{['eq:inversion_map']}, and shown in the left column. The mesh of the inverse domain is shown in the center. An iso-lines representation of the function evaluated in a grid of size $(1\,000 \times 1\,000)$ within the cube $[-3, 3]$ is shown in the right column.
• Figure 4: Comparison of query time statistics as a function of the mesh resolution parameter $l_x$, and the usage of a hint (warm start). In continuous motion, the hint can be set as the previous result (enclosing cell or tetra), and leads to query times considerably under a microsecond.
• Figure 5: We visualize the semantics-aware distance of a combination wrench through its iso-surfaces, at three different semantic configurations. The striped surface regions represent the surface regions where contact is allowed, i.e., where the zero level-set of the computed function lies within the object.