On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis (2302.10433v3)

Abstract: We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group $\G$ and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and $\G$-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.

• The paper proposes a group-theoretic framework to identify discrete morphological symmetries in robotics systems, applicable to various locomotion mechanisms.
• It demonstrates that system symmetries extend to dynamics and control strategies, enabling more efficient and optimal system functions.
• Utilizing symmetries as an inductive bias enhances data-driven methods, improving sample efficiency and generalization in learning for robotics tasks.

On Discrete Symmetries of Robotics Systems: A Group-Theoretic and Data-Driven Analysis

The paper "On Discrete Symmetries of Robotics Systems: A Group-Theoretic and Data-Driven Analysis" by Daniel Ordonez-Apraez et al. explores discrete morphological symmetries (DMS) in robotics systems through a comprehensive group-theoretic and data-driven approach. This paper is pivotal for identifying inherent symmetries in both biological and artificial locomotion mechanisms, such as legged, swimming, and flying animals/robots.

Key Contributions

1. Theoretical Framework for Discrete Morphological Symmetries: The authors propose a theoretical framework predicated on the concepts of group theory to identify and characterize DMSs in dynamical systems. They adapt transformations such as rotations, reflections, and translations to describe repetitions or reflections in the geometrical structure of a system. These are considered within the symmetry group $G$, allowing systems to imitate Euclidean isometries through discrete morphological modifications.
2. Significance in Dynamics and Control: A novel insight from the paper is the idea that symmetries in system morphology extend to dynamic processes, including control strategies and sensory data. This equivalence suggests that dynamic and control functions mirror the system’s symmetrical properties, facilitating optimal strategies that preserve the intrinsic balance of the system.
3. Application in Data-Driven Methodologies: By incorporating symmetry as an inductive bias, the paper leverages data augmentation and $G$-equivariant neural networks to enhance sample efficiency and generalization capabilities. The authors effectively demonstrate that acknowledging symmetries can reduce trainable parameters and improve learning processes for robotics and related applications.
4. Practical and Computational Implementations: An open-source repository introduced by the authors exemplifies the practical utilization of their framework, alongside tools for the implementation of $G$-equivariant networks. This aspect facilitates adaptable experiments that focus on identifying symmetry groups within large-scale data and controlling the complexities associated with neural network architectures.

Experimental Evidence

The paper features rigorous evidence through experiments on synthetic and real-world scenarios. The experimentation demonstrates several numerical advantages of applying the authors' approach, such as increased efficiency in training data and reduced model overfitting. Particularly in the tasks of center of mass momentum estimation and contact state detection, the models exemplified enhanced performance under symmetric exploitation compared to conventional models.

Implications and Future Directions

This research bears significant practical implications. The structured identification and utilization of symmetry could lead to the development of more robust control algorithms, potentially enhancing robotic systems’ adaptability and resilience to physical perturbations or environmental changes. In a theoretical context, it broadens our understanding of symmetry integration within modern machine learning paradigms, promising new avenues for algorithmic efficiency.

Looking forward, one prominent area for further exploration could involve catering to approximate symmetries in more realistic settings since exact symmetry conditions are an idealization. Moreover, investigating the impacts of dynamically evolving symmetry groups in adaptive systems could be a worthwhile pursuit. Extending this work to capture continuous groups beyond discrete settings may unveil broader applications across other domains dealing with morphologically intricate systems.

In sum, this paper provides a robust platform, both theoretical and computational, for leveraging symmetries in dynamically complex systems, with notable implications for the evolution of robotics both in design and operational capacity.