Ensuring symmetric stiffness in impedance-controlled robots is crucial for physically meaningful and stable interaction in contact-rich manipulation. Conventional approaches neglect the change of basis vectors in curved spaces, leading to an asymmetric joint-space stiffness matrix that violates passivity and conservation principles. In this work, we derive a physically consistent, symmetric joint-space stiffness formulation directly from the task-space stiffness matrix by explicitly incorporating Christoffel symbols. This correction resolves long-standing inconsistencies in stiffness modeling, ensuring energy conservation and stability. We validate our approach experimentally on a robotic system, demonstrating that omitting these correction terms results in significant asymmetric stiffness errors. Our findings bridge theoretical insights with practical control applications, offering a robust framework for stable and interpretable robotic interactions.

翻译：确保阻抗控制机器人具有对称刚度对于接触密集型操作中物理意义明确且稳定的交互至关重要。传统方法忽略了弯曲空间中基向量的变化，导致关节空间刚度矩阵不对称，违反了无源性和守恒原理。在本研究中，我们通过显式引入克里斯托费尔符号，直接从任务空间刚度矩阵推导出物理一致、对称的关节空间刚度公式。该修正解决了刚度建模中长期存在的不一致问题，确保了能量守恒与稳定性。我们在机器人系统上通过实验验证了该方法，证明忽略这些修正项会导致显著的非对称刚度误差。我们的研究将理论洞见与实际控制应用相结合，为稳定且可解释的机器人交互提供了一个鲁棒框架。