Attention-Based Surface Representation Learning for Robot State Prediction and Open-Ended Surface Classification

All the objects composing our world are enclosed within surfaces. Yet, most robot learning and motion generation frameworks treat surfaces as constraints ignoring their intrinsic geometry. This gap is acute for polyhedral meshes--the standard output of CAD and 3D reconstruction--whose discrete geometric structure remains unexploited. In this paper, we propose a unified discrete Riemannian framework that enables robot learning directly on polyhedral surface meshes. Using discrete differential geometry, we define logarithmic and exponential maps, parallel transport, and ambient-space projections that remain well-defined across faces, edges, and vertices. We instantiate the framework in three learning paradigms: (i) Dynamic Movement Primitives (DMPs), an improved exponential-map computation and a fixed-tangent-cone forcing-term encoding with parallel transport yield better cross-surface generalisation and stability over prior mesh-based approaches. (ii) Gaussian Process (GP), a geodesic-based kernel with practical admissibility control, enables regression at arbitrary mesh locations without smoothness assumptions. (iii) Riemannian Flow Matching (RFM), mesh-native operators improve generative quality over spectral baselines while reducing training time. The framework is validated in simulation against state-of-the-art methods and demonstrated on two real-robot scenarios: generalising user-drawn trajectories across different surfaces and planning polishing motions on RGB-D-reconstructed surfaces.

The energy-based method remains a comparatively underexamined approach for surface classification in mobile robotics, despite promising results in constrained environments. This study evaluated the viability of using energy-derived features as either a standalone classification modality or as supplementary input to inertial data. A comprehensive evaluation was conducted across three publicly available datasets, comparing the performance of modern deep learning architectures including recurrent neural networks, convolutional neural networks, encoder-only transformers, and Mamba state-space models, under automated hyperparameter tuning and input sequence length optimization. The models achieved higher accuracy than previously reported values on all evaluated datasets, with the convolutional neural network yielding the highest overall performance. When relying exclusively on energy-based features, the models attained classification accuracies in the range of 85-90%, approximately 5-10% lower than those achieved when combined with inertial features (96-99%). Augmenting inertial data with energy features resulted in a consistent mean accuracy improvement of 1-2%. These findings indicate that classifiers relying solely on energy features offer sufficient accuracy for standalone deployment, while also providing a consistent gain when used in combination with other sensing modalities.

World models allow robots to anticipate action consequences before execution. This capability is especially valuable in excavation, where each scoop reshapes the terrain and affects subsequent actions. Local observations, however, cannot fully reveal the underlying support and material conditions. We present PileBelief, an interaction-driven persistent world model for partially observed excavation that retains physical evidence beyond the visible surface. It combines an observation-conditioned physical prior with world-addressed deformation memory and physical-response memory. Action-aligned reads and gated residual corrections refine terrain-change and outcome predictions. With deployment weights fixed, completed interactions update measured belief, while hypothetical actions advance a separate imagined state. Compared with a current-observation-only baseline, PileBelief reduces five-step joint prediction error by 10.8% and offline action-selection regret by 65.5%. Experiments on Newton/MPM and real excavation datasets further demonstrate improved terrain-change and bucket-volume prediction. Our method enables multi-step prediction and candidate-action ranking from local observations, even when the underlying soil state is unknown. These results identify persistent physical belief as a useful representation for world models of environments that robots continually reshape.