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 system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, demonstrates enhanced generalization, sample efficiency, and interpretability, with less trainable parameters and computational costs.

翻译：我们引入谐波分析来将对称机器人系统的状态空间分解为正交等型子空间。这些低维空间能够捕捉不同且对称协同的运动模式。针对线性动力学系统，我们阐明了这种分解如何导致动力学在每个子空间上被划分为独立的线性系统，我们将此性质定义为动力学谐波分析。为利用该性质，我们基于库普曼算子理论提出了一种等变深度学习架构，该架构借助动力学谐波分析的特性学习系统动力学的全局线性模型。在合成系统及四足机器人运动动力学上的验证表明，该架构在减少可训练参数和计算成本的同时，显著提升了泛化能力、样本效率与可解释性。