Geometric Formulation of Unified Force-Impedance Control on SE(3) for Robotic Manipulators

In this paper, we present an impedance control framework on the SE(3) manifold, which enables force tracking while guaranteeing passivity. Building upon the unified force-impedance control (UFIC) and our previous work on geometric impedance control (GIC), we develop the geometric unified force impedance control (GUFIC) to account for the SE(3) manifold structure in the controller formulation using a differential geometric perspective. As in the case of the UFIC, the GUFIC utilizes energy tank augmentation for both force-tracking and impedance control to guarantee the manipulator's passivity relative to external forces. This ensures that the end effector maintains safe contact interaction with uncertain environments and tracks a desired interaction force. Moreover, we resolve a non-causal implementation problem in the UFIC formulation by introducing velocity and force fields. Due to its formulation on SE(3), the proposed GUFIC inherits the desirable SE(3) invariance and equivariance properties of the GIC, which helps increase sample efficiency in machine learning applications where a learning algorithm is incorporated into the control law. The proposed control law is validated in a simulation environment under scenarios requiring tracking an SE(3) trajectory, incorporating both position and orientation, while exerting a force on a surface. The codes are available at https://github.com/Joohwan-Seo/GUFIC_mujoco.

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

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.