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

Daniel Ordonez-Apraez, Mario Martin, Antonio Agudo, Francesc Moreno-Noguer

Keywords: Morphological Symmetries, Discrete Symmetries of Dynamical Systems, Equivariant Dynamics, Equivariant Function Approximators, Geometric Deep Learning

TL;DR: We present a group-theoretic analysis of bilateral/radial symmetries of dynamical systems. Characterizing the symmetries of the system's dynamics, control, and proprioceptive/exteroceptive data. And elucidating how to exploit these symmetries in DL

Abstract: In this work, we study the Morphological Symmetries of dynamical systems with one or more planes of symmetry, a predominant feature in animal biology and robotic systems, characterized by 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) the characterization of how the group acts upon the system state variables and any relevant measurement living in the Euclidean space, and (3) the exploitation of data symmetries through the use of $\G$-equivariant/$\G$-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.

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