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.

翻译：机器人因其机械结构及任务特性而展现出丰富的对称性。尽管许多机器人学问题同时存在多重对称性，但现有方法通常孤立处理它们，未能充分利用其组合潜力。本文提出跨空间对称组合框架，用于学习在构型空间与任务空间中同时对多重对称性具有等变性的机器人策略。通过利用正向运动学映射的微分几何结构，我们将对称性从构型空间下降至任务空间，并从任务空间提升至构型空间，从而在统一表示空间内实现其组合。我们在双机械臂的仿真与真实世界实验中验证了该框架，结果表明联合利用多重对称性能够提升泛化性能。