SE(3)-Equivariant Robot Learning and Control: A Tutorial Survey (2503.09829v3)

Abstract: Recent advances in deep learning and Transformers have driven major breakthroughs in robotics by employing techniques such as imitation learning, reinforcement learning, and LLM-based multimodal perception and decision-making. However, conventional deep learning and Transformer models often struggle to process data with inherent symmetries and invariances, typically relying on large datasets or extensive data augmentation. Equivariant neural networks overcome these limitations by explicitly integrating symmetry and invariance into their architectures, leading to improved efficiency and generalization. This tutorial survey reviews a wide range of equivariant deep learning and control methods for robotics, from classic to state-of-the-art, with a focus on SE(3)-equivariant models that leverage the natural 3D rotational and translational symmetries in visual robotic manipulation and control design. Using unified mathematical notation, we begin by reviewing key concepts from group theory, along with matrix Lie groups and Lie algebras. We then introduce foundational group-equivariant neural network design and show how the group-equivariance can be obtained through their structure. Next, we discuss the applications of SE(3)-equivariant neural networks in robotics in terms of imitation learning and reinforcement learning. The SE(3)-equivariant control design is also reviewed from the perspective of geometric control. Finally, we highlight the challenges and future directions of equivariant methods in developing more robust, sample-efficient, and multi-modal real-world robotic systems.

An Overview of $SE(3)$-Equivariant Robot Learning and Control

This paper offers a tutorial survey on the development of $SE(3)$-equivariant models within the domain of robotic learning and control. The focus lies on how these models leverage natural symmetries inherent in space to enhance the efficiency and generalization capabilities of robotic systems.

Symmetry and Invariance in Robotic Learning

Robotic learning has benefited from recent advancements in deep learning, where models like Transformers enable robots to interpret and adapt within complex environments. However, conventional models fall short when handling data characterized by symmetries and invariances. Such deficiencies typically necessitate large datasets and extensive data augmentation to improve model performance. To overcome these constraints, equivariant neural networks have emerged, integrating symmetry directly into their architectures to address these challenges. This approach allows the models to exploit the $SE(3)$ symmetries, representing 3D rotational and translational transformations, central to the control and manipulation tasks in robotics.

Progress in Equivariant Models

The paper systematically reviews a broad spectrum of equivariant modeling techniques, spanning from foundational concepts to state-of-the-art implementations. It initiates with principles from group theory that form the backbone of these models, specifically those grounded in $SE(3)$ manifolds. Matrix Lie groups and algebras are discussed, emphasizing their relevance to understanding and designing group-equivariant neural networks.

One critical component of these networks is the incorporation of symmetry into neural network architectures, ensuring that transformations in inputs result in predictable transformations in outputs. The survey covers $SE(3)$-equivariant models, which are particularly suited to processing visual data in robotic manipulation through techniques such as imitation and reinforcement learning. These models enhance learning efficiency by recognizing and leveraging the natural symmetry present in visual data inputs.

Equivariant Control Design

Additionally, the paper explores control strategies within the $SE(3)$ framework. It examines how equivariance principles can be applied at both force and control levels to achieve more effective manipulation behaviors in robotics. The key lies in using geometric control approaches that align with the manifold structure of $SE(3)$. By employing $SE(3)$ manifolds, robots can operate using control laws that exploit natural symmetries in motion and force applications, enhancing robustness and efficiency.

Challenges and Future Directions

This tutorial highlights several challenges in the implementation of equivariant models, such as the need for more robust models capable of navigating dynamic environments and integrating data from multiple modalities. The paper posits potential future advancements in multi-modal sensor fusion and lifelong learning for robotic systems, driven by the continued evolution of equivariant methodologies that harness the $SE(3)$ symmetries.

Thus, this survey serves as an insightful roadmap for experienced researchers endeavoring into the field of $SE(3)$ models, offering a comprehensive overview that bridges theoretical constructs to practical implementations, setting the stage for future breakthroughs in more sophisticated and adaptable robotic systems.