  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.
