We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of the system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, exhibits enhanced generalization, sample efficiency, and interpretability, with fewer trainable parameters and computational costs.

翻译：本文引入谐波分析方法，将对称机器人系统的状态空间分解为正交同型子空间。这些低维空间能够捕捉独特、对称且协同的运动模式。针对线性动力学，我们阐述了这种分解如何将动力学细分为各子空间上的独立线性系统，该特性我们称之为动态谐波分析（DHA）。为利用此特性，我们结合Koopman算子理论，提出一种等变深度学习架构，该架构借助DHA的特性来学习系统动力学的全局线性模型。我们在合成系统与四足机器人运动动力学上的验证表明，该架构具有更强的泛化能力、样本效率和可解释性，同时所需可训练参数和计算成本更低。