On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis

In this work, we study discrete morphological symmetries of dynamical systems, a predominant feature in animal biology and robotic systems, expressed when the system's morphology has one or more planes of symmetry describing the duplication and balanced distribution of body parts. These morphological symmetries imply that the system's dynamics are symmetric (or approximately symmetric), which in turn imprints symmetries in optimal control policies and in all proprioceptive and exteroceptive measurements related to the evolution of the system's dynamics. For data-driven methods, symmetry represents an inductive bias that justifies data augmentation and the construction of symmetric function approximators. To this end, we use group theory to present a theoretical and practical framework allowing for (1) the identification of the system's morphological symmetry group $\G$, (2) data-augmentation of proprioceptive and exteroceptive measurements, and (3) the exploitation of data symmetries through the use of $\G$-equivariant/invariant neural networks, for which we present experimental results on synthetic and real-world applications, demonstrating how symmetry constraints lead to better sample efficiency and generalization while reducing the number of trainable parameters.