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: ❈ A group-theoretic formalization of the concept of discrete morphological symmetry. ❈ A characterization of how the morphological symmetry group G affects the system's state variables and any relevant proprioceptive and exteroceptive measurements. Facilitating the identification of G and the augmentation of proprioceptive and exteroceptive measurements. f 1 gr • f 1 g t • f 1 gs • f 1 e gr gt gs g t gs gs g t g r h . = l k gs • h g t • h gr • h g • y = f (g • x; ϕ) | ∀ g ∈ K 4 ϕ . = ¶ 0 c, . . . , l c © K 4 = ¶ e, gs, g t , gr | g 2 s = g 2 t = g 2 r = e, gr = gsg t © 0 c + 0 W × σ l c + × σ f (x; ϕ) l W 0 B l B x gs • x g t • x gr • x 0 z l- Once the morphological symmetry group G is identified, our practical contributions focus on the efficient construction and versatile use of G-equivariant neural networks, for arbitrary discrete morphological symmetry groups G, for which we: ✥ Derived an optimal initialization for the trainable parameters of equivariant layersfoot_1 . ✥ Demonstrate that G-equivariance reduces the trainable parameters by approximately 1 /|G|. ✥ Enable the construction of large scale G-equivariant networks by mitigating the construction computational complexity and the storage memory complexity of equivariant architectures 2 .

2. BACKGROUND ON SYMMETRY GROUPS

In a nutshell, a symmetry group in Group Theory is an abstraction of the concept of symmetries that different geometric objects might have, understanding symmetry as a transformation that when applied to an object conserves a relevant property of its structure. For instance, in fig. 1 -left the Klein four-group K 4 describes the symmetries that vectors, pseudo-vectors, rigid bodies, and a quadruped robot have to 180 • rotations (g r ) and two perpendicular reflections (g s , g t ). Transformations that preserve vector magnitudes and energy. While on fig. 1 -right the same group describes the symmetries of vector spaces, representing the quadruped robot's state x and legs contact state y. Formally, a symmetry group is a set of invertible symmetry transformations (or actions) G = {e, g 1 , g 2 , . . . }, containing the trivial action e (which leaves objects unchanged) and having a binary operator ( •) : G × G → G, that is associative (i.e. g 1 • (g 2 • g 3 ) = (g 1 • g 2 ) • g 3 ), which composes group members into other group members, such as g r = g s • g t for K 4 (see fig. 1 ). Group representations are characterizations of how each action g transforms a specific geometric object, say x ∈ R k . A representation ρ x : G → GL(k) (GL : General Linear group) is a group homomorphism associating each g to an invertible linear map ρ x (g) ∈ R k×k specifying how the object x is transformed, that is: g(x) ≡ g • x . = ρ x (g)x. Since group actions are abstract, it is common to define different object-dependent representations for each action, as we will see throughout this work.



The field of mathematics that studies symmetries, which is broadly used in Machine Learning(ML) The link to an anonymous repository is available in the official comments on OpenReview, accessible to reviewers and area chairs. The repository will be made public to the general public upon paper acceptance.



Figure 1: Left: Caley diagram and top-view (see 3D animation) of symmetric configurations of the quadruped robot Solo, whose morphological symmetries (described by the Klein four-group K 4 ) allow it to imitate the effect of reflections (g s , g t ) and 180 • rotations of space (g r ). Transformations affect both proprioceptive (state space, CoM linear l and angular k momentum) and exteroceptive (terrain elevation, external disturbances) quantities. Right: Diagram of a K 4 -equivariant NN. Each of the layer's linear maps W is constructed as a weighted average of the basis of the space of equivariant linear maps B, computed from the K 4 symmetries of the input-output spaces (see section 5).

