Symmetries Here and There, Combined Everywhere: Cross-space Symmetry Compositions in Robotics

2026-05-21

AI Summary

Existing robotic learning approaches typically handle symmetries in configuration space and task space in isolation, failing to exploit their combined benefits. This work proposes a cross-space symmetry composition framework that, grounded in the differential geometric structure of forward kinematics, enables for the first time bidirectional transfer, unified integration, and equivariant policy learning across both symmetry types. By introducing geometry-driven symmetry lifting and reduction mechanisms, the method constructs joint equivariant representations that significantly enhance policy generalization, as demonstrated in both simulated and real-world dual-arm robot experiments. This approach transcends conventional modeling paradigms that rely on single-space symmetries alone.

Abstract

Robots exhibit a rich variety of symmetries arising from their mechanical structure and the properties of their tasks. Although many robotics problems exhibit several symmetries simultaneously, existing approaches typically treat them in isolation, failing to exploit their combined potential. This paper introduces cross-space symmetry compositions, a framework for learning robot policies that are jointly equivariant to multiple symmetries across configuration and task spaces. Leveraging the differential-geometric structure of the forward kinematics map, we both descend symmetries from configuration to task space and lift symmetries from task to configuration space, enabling their composition within a unified representation space. We validate our framework on simulated and real-world experiments on a dual-arm robot, demonstrating that jointly leveraging multiple symmetries yields improved generalization.

Problem

• symmetry
• robotics
• equivariance
• generalization
• cross-space

Innovation

• cross-space symmetry
• equivariant policy
• differential geometry
• forward kinematics
• symmetry composition