ErgoSurf: Ergodic Control for the Coverage of Unknown Surfaces

Stefan Schneyer, Timo Bachmann, Maged Iskandar, Korbinian Nottensteiner, Alin Albu-Schäffer, Freek Stulp, João Silvério · N/A · 2026

Summary

Contact-centric tasks on surfaces, ranging from inspection and cleaning to sanding and polishing, require robots to systematically cover the surface while maintaining stable contact.

Abstract Summary

Contact-centric tasks on surfaces, ranging from inspection and cleaning to sanding and polishing, require robots to systematically cover the surface while maintaining stable contact. Ergodic control generates trajectories that spend time at a location proportional to a desired, task-specific spatial distribution, enabling efficient information gathering and coverage. However, traditional ergodic control methods rely on prior knowledge of surface geometry or require a vision sensory input to scan the geometry beforehand, limiting their applicability in real-world scenarios with unknown or dynamic environments. This paper introduces a novel online ergodic control framework that achieves systematic surface coverage while simultaneously reconstructing unknown surface geometry. We employ a Gaussian Process Implicit Surface (GPIS) model that learns global surface geometry from intrinsic tactile sensing during execution. For efficient online planning, we approximate the surface locally using point clouds sampled from tangent planes at observed contact points and iteratively fit them to the Gaussian Process. This approximation simultaneously serves as the sampling domain for both the target and the coverage distributions. We employ a heat-diffusion analogy to compute potential fields that guide ergodic exploration, translating spatial coverage objectives into smooth robot trajectories. We demonstrate our framework through simulation and real-robot experiments, validating simultaneous ergodic coverage and online surface geometry learning with reconstruction errors approaching the ground truth.

Key Points

Contact-centric tasks on surfaces, ranging from inspection and cleaning to sanding and polishing, require robots to systematically cover the surface while maintaining stable contact.
Ergodic control generates trajectories that spend time at a location proportional to a desired, task-specific spatial distribution, enabling efficient information gathering and coverage.
However, traditional ergodic control methods rely on prior knowledge of surface geometry or require a vision sensory input to scan the geometry beforehand, limiting their applicability in real-world scenarios with unknown or dynamic environments.
This paper introduces a novel online ergodic control framework that achieves systematic surface coverage while simultaneously reconstructing unknown surface geometry.
We employ a Gaussian Process Implicit Surface (GPIS) model that learns global surface geometry from intrinsic tactile sensing during execution.

Proposed Approach

This section presents ErgoSurf, our ergodic control approach on surfaces with unknown geometry. Let $\Omega\subset\mathbb{R}^{3}$ denote the domain space of interest (see Fig. 3 ), and let the agent’s task space be represented as the surface set $\mathcal{S}\subset\Omega$.

For in-contact tasks, $\mathcal{S}$ comprises all feasible contact points within $\Omega$, constraining the robot’s motion to this subset during environment interaction.

Proposed Approach — ErgoSurf: Ergodic Control for the Coverage of Unknown Surfaces — Figure 4: Tangent plane sampling at a contact point $\bm{q}_{k}$ (green) on the current GPIS zero-level set estimate (gray surface). The GPIS normal $\hat{\bm{n}}_{k}$ defines the local tangent plane $\Pi_{k}$ with orthonormal basis $\mathbf{T}_{k}=\left[\bm{\tau}_{1},\bm{\tau}_{2}\right]$. Points $\bm{z}$ are drawn from a Gaussian in the tangent plane ( 15 ) (spread $\sigma_{\text{tp}}$ shown as the circle) and collected in the point cloud $\mathcal{Z}$.

Experiments and Results

We validate the proposed framework through simulation and real-world experiments with a single agent, demonstrating simultaneous surface reconstruction and ergodic coverage of unknown surfaces without prior geometric knowledge. The bunny, chair, and backpanel experiments are shown in the supplementary video. We assess performance using three different metrics across the experiments.

The ergodic cost $\epsilon_{k}$ ( 10 ) quantifies coverage quality relative to the target distribution.

Experiments and Results — ErgoSurf: Ergodic Control for the Coverage of Unknown Surfaces — Figure 5: Projection refinement of the tangent plane samples. One refinement step moves each point $\bm{z}_{m}$ from the tangent plane $\Pi_{k}$ ( Fig. 4 ) toward the current GPIS zero-level set estimate along the estimated normal by the signed distance ( 16 ), giving the updated point $\bm{z}_{m+1}$; the step is iterated until the point reaches the surface. Corrections vanish near the contact $\bm{q}_{k}$, where the tangent plane already matches the surface, and grow with distance as curvature bends the surface away from the plane.

Sources and Demonstrations

Figures are reproduced from Schneyer et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.