Dynamical Systems (DS) are an effective and powerful means of shaping high-level policies for robotics control. They provide robust and reactive control while ensuring the stability of the driving vector field. The increasing complexity of real-world scenarios necessitates DS with a higher degree of non-linearity, along with the ability to adapt to potential changes in environmental conditions, such as obstacles. Current learning strategies for DSs often involve a trade-off, sacrificing either stability guarantees or offline computational efficiency in order to enhance the capabilities of the learned DS. Online local adaptation to environmental changes is either not taken into consideration or treated as a separate problem. In this paper, our objective is to introduce a method that enhances the complexity of the learned DS without compromising efficiency during training or stability guarantees. Furthermore, we aim to provide a unified approach for seamlessly integrating the initially learned DS's non-linearity with any local non-linearities that may arise due to changes in the environment. We propose a geometrical approach to learn asymptotically stable non-linear DS for robotics control. Each DS is modeled as a harmonic damped oscillator on a latent manifold. By learning the manifold's Euclidean embedded representation, our approach encodes the non-linearity of the DS within the curvature of the space. Having an explicit embedded representation of the manifold allows us to showcase obstacle avoidance by directly inducing local deformations of the space. We demonstrate the effectiveness of our methodology through two scenarios: first, the 2D learning of synthetic vector fields, and second, the learning of 3D robotic end-effector motions in real-world settings.

翻译：动力系统（DS）是机器人控制中制定高层策略的有效且强大的手段。它们在确保驱动向量场稳定性的同时，提供鲁棒且反应式控制。现实世界场景日益增长的复杂性，要求DS具有更高的非线性程度，并具备适应潜在环境变化（如障碍物）的能力。当前DS的学习策略往往需要权衡：要么牺牲稳定性保证，要么牺牲离线计算效率，以增强所学DS的能力。对环境变化的在线局部适应要么未被考虑，要么被作为独立问题处理。本文旨在提出一种方法，在不牺牲训练效率或稳定性保证的前提下，增强所学DS的复杂性。此外，我们力求提供一种统一方法，将初始学习的DS非线性与因环境变化可能产生的局部非线性无缝集成。我们提出一种几何方法，用于学习渐近稳定的非线性DS以实现机器人控制。每个DS被建模为潜在流形上的谐波阻尼振荡器。通过学习流形的欧几里得嵌入表示，我们的方法将DS的非线性编码于空间曲率之中。拥有流形的显式嵌入表示使我们能够通过直接引入空间的局部形变来演示避障。我们通过两个场景证明了该方法有效性：一是合成向量场的二维学习，二是真实世界环境下三维机器人末端执行器运动的学习。