ON DISCRETE SYMMETRIES OF ROBOTICS SYSTEMS: A GROUP-THEORETIC AND DATA-DRIVEN ANALYSIS Anonymous

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 proprioceptive and exteroceptive measurement, 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.

1. INTRODUCTION

Symmetries are a predominant feature in animal biology. The majority of living (and extinct) species are bilaterally or radially symmetric (i.e., having one or more planes of symmetry), a property intuitively recognized by the patterns of balanced distribution and duplication of body parts and shapes (Holló, 2017) . Likewise, most robotic systems are symmetric, often featuring more precise symmetries than nature due to the accurate duplication of body parts and the tendency to design mechanisms with symmetric volumes and mass distributions. These morphological symmetries of animals and robots imply that the dynamics and control of body motions are also approximately symmetric, resulting in all proprioceptive and exteroceptive measurements, related to the evolution of the system's dynamics (e.g. joint torques, depth images, contact forces), to be also symmetric. This highly relevant inductive bias is frequently left unexploited in most data-driven applications in the fields of robotics, computer graphics, computational biology, and control. Recent works in computer graphics (Yeh et al., 2019; Abdolhosseini et al., 2019; Yu et al., 2018) and robotics/dynamical systems ( Van der Pol et al., 2020; Ordonez-Apraez et al., 2022; Hamed & Grizzle, 2013; Finzi et al., 2021a) have exploited through different approaches the morphological symmetry group associated with bilateral (or sagittal) symmetry (the reflection group C 2 ), obtaining improvements in generalization and sample efficiency of function approximators. Notably, Zinkevich & Balch (2001) proved that Markov Decision Processes with state symmetries have symmetric optimal value and policy functions. Despite these encouraging contributions, exploiting the inductive bias of morphological symmetries is not a widespread technique in the research community. We attribute the scarce adoption of these techniques to the lack of a unifying theoretical and practical framework, allowing to identify different morphological symmetries in arbitrary dynamical systems and efficiently and conveniently exploit them in data-driven applications. This work takes a step towards this unifying framework by studying morphological symmetries through the lens of dynamical systems and group theoryfoot_0 . Our theoretical contributions are:



The field of mathematics that studies symmetries, which is broadly used in Machine Learning (ML) 1

