发表机构
Università di Trento; New York University Abu Dhabi（特伦托大学; 纽约大学阿布扎比分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个统一的离散黎曼框架，使机器人学习直接在多面体表面网格上进行，通过离散微分几何定义映射和平行移动，并在DMP、GP和RFM三种范式中验证，实现跨表面泛化和真实机器人应用。
AI 中文摘要
构成我们世界的所有物体都被表面所包围。然而，大多数机器人学习和运动生成框架将表面视为约束，而忽略了其内在几何结构。对于多面体网格——CAD和三维重建的标准输出——这一差距尤为严重，其离散几何结构仍未得到利用。在本文中，我们提出一个统一的离散黎曼框架，使机器人学习能够直接在多面体表面网格上进行。利用离散微分几何，我们定义了对数映射和指数映射、平行移动以及环境空间投影，这些在面、边和顶点上均保持良好定义。我们在三种学习范式中实例化该框架：（i）动态运动原语（DMPs），改进的指数映射计算和带有平行移动的固定切锥强迫项编码，相比先前的基于网格的方法，实现了更好的跨表面泛化和稳定性。（ii）高斯过程（GP），基于测地线的核并具有实用的可容许性控制，能够在无需平滑假设的情况下在任意网格位置进行回归。（iii）黎曼流匹配（RFM），网格原生算子相比谱基线提高了生成质量，同时减少了训练时间。该框架在仿真中与最先进方法进行了验证，并在两个真实机器人场景中进行了演示：跨不同表面泛化用户绘制的轨迹，以及在RGB-D重建表面上规划抛光运动。
英文摘要
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.
